side above is equivalent to the right hand side of (7.8); this follows by
Lemma 2.3. In other words, (7.8) and (7.9) agree. ∎

Recall that the spectra $K(ku_{p})_{i}$ are given in Construction 7.3 and the
groups $V(1)_{*}K(ku_{p})_{i}$ are identified in Theorem 7.5.

######  Theorem 7.10.

For $p>3$, there is an equivalence of spectra:

$K(ko_{p})\simeq\bigvee_{0\leq i<(p-1)/2}K(ku_{p})_{2i}.$

Therefore, we have

$V(1)_{*}K(ko_{p})\cong\bigoplus_{0\leq i<(p-1)/2}V(1)_{*}K(ku_{p})_{2i}.$

and $V(1)_{*}K(ko_{p})$, as an abelian group, is given by:

$\begin{split}V(1)_{*}K(ko_{p})\cong&\mathbb{F}_{p}[b^{2}]\otimes\Lambda(\lambda_{1},ba_{1})\oplus\mathbb{F}_{p}[b^{2}]\otimes\mathbb{F}_{p}\\{\partial\lambda_{1},b\partial
b,b\partial a_{1},b\partial\lambda_{1}a_{1}\\}\\\
&\oplus\mathbb{F}_{p}[b^{2}]\otimes\Lambda(ba_{1})\otimes\mathbb{F}_{p}\\{t^{d}\lambda_{1}\mid
0<d<p\\}\\\
&\oplus\mathbb{F}_{p}[b^{2}]\otimes\Lambda(\lambda_{1})\otimes\mathbb{F}_{p}\\{b^{\epsilon(n)}\sigma_{n},\lambda_{2}t^{p^{2}-p}\mid
1\leq n\leq p-2\\}\\\ &\oplus\mathbb{F}_{p}\\{s\\},\end{split}$

where $\epsilon(n)=1$ if $n$ is odd and $\epsilon(n)=0$ if $n$ is even. Here,
the class denoted by $(b^{2})^{(p-1)/2}$ is $-v_{2}$.

As a consequence, we have an isomorphism of abelian groups:

$T(2)_{*}K(ko)\cong T(2)_{*}K(\ell_{p})[b^{2}]/((b^{2})^{(p-1)/2}+v_{2}).$

###### Proof.

We start by identifying $\operatorname{\textup{THH}}(ko_{p})$ as a cyclotomic
spectrum. We have the following chain of equivalences

$\begin{split}\operatorname{\textup{THH}}(ko_{p})&\simeq\operatorname{\textup{THH}}(ko_{p}(\sqrt[2]{\alpha}))_{0}\\\
&\simeq\operatorname{\textup{THH}}(D^{\prime}(ku_{p}))_{0}\\\
&\simeq\big{(}D^{\prime}(\operatorname{\textup{THH}}(ku_{p}))\big{)}_{0}\\\
&\simeq\prod_{0\leq
i<(p-1)/2}\operatorname{\textup{THH}}(ku_{p})_{2i}\end{split}$

The first equivalence above follows by Theorem 4.20, the second equivalence
follows by Proposition 7.7 and the third equivalence is a consequence of [2,
Corollary A.15]. The last equivalence above follows by the description of
$D^{\prime}$ as a left Kan extension, see Section 2.2. Indeed, this shows that
the following composite map of cyclotomic spectra is an equivalence.

$\operatorname{\textup{THH}}(ko_{p})\to\operatorname{\textup{THH}}(ku_{p})\simeq\prod_{0\leq
i<p-1}\operatorname{\textup{THH}}(ku_{p})_{i}\to\prod_{0\leq
i<(p-1)/2}\operatorname{\textup{THH}}(ku_{p})_{2i}$

Here, the equivalence in the middle follows by Proposition 7.2. The last map
above is the canonical projection.

The composite equivalence of cyclotomic spectra above shows that

$\operatorname{\textup{TC}}(\operatorname{\textup{THH}}(ko_{p}))\simeq\operatorname{\textup{TC}}(\prod_{0\leq
i<(p-1)/2}\operatorname{\textup{THH}}(ku_{p})_{2i})\simeq\prod_{0\leq
i<(p-1)/2}\operatorname{\textup{TC}}(ku_{p})_{2i}.$

Considering the Dundas-Goodwillie-McCarthy theorem with respect to the
composite $ko_{p}\to ku_{p}\to H\mathbb{Z}_{p}$, we obtain that the splitting
of the pullback square relating $K(ku_{p})$ with
$\operatorname{\textup{TC}}(ku_{p})$ (mentioned in Construction 7.3) provides
a splitting for $K(ko_{p})$ given by

(7.11) $K(ko_{p})\simeq\prod_{0\leq i<(p-1)/2}K(ku_{p})_{2i}.$

The first and the second statements in the theorem follow from this splitting.
The third statement follows by this, and by inspection on (7.4).

For the last statement, note that $T(2)_{*}K(ko)\cong T(2)_{*}K(ko_{p})$ due
to the purity of algebraic $K$-theory and [31, Lemma 2.2 (vi)]. It follows by
Theorem 7.5, that the map

$T(2)_{*}K(ku_{p})\xrightarrow{\cdot b^{i}}T(2)_{*}K(ku_{p})$

carries $T(2)_{*}K(ku_{p})_{0}$ to $T(2)_{*}K(ku_{p})_{i}$ for $i<p-1$ where
the map above multiplies by $b^{i}$. Using this fact, together with [7,
Proposition 1.2 (b)], provides isomorphisms

$T(2)_{*}K(\ell_{p})\cong
T(2)_{*}K(ku_{p})_{0}\xrightarrow{\cong}T(2)_{*}K(ku_{p})_{i}$

given by $\cdot b^{i}$ for $i<p-1$. This, together with (7.11) provides the
desired identification of $T(2)_{*}K(ko)\cong T(2)_{*}K(ko_{p})$ as
$T(2)_{*}K(\ell_{p})[b^{2}]/((b^{2})^{(p-1)/2}+v_{2})$. ∎

##  8\. Root adjunction and Lubin-Tate spectra

Recall that in [25], Hahn and Wilson prove that there are $E_{3}$ $MU$-algebra
forms of $BP\langle n\rangle$. Furthermore, their constructions provide an
$E_{3}$ $MU[\sigma_{2(p^{n}-1)}]$-algebra form of $BP\langle n\rangle$ where
$\sigma_{2(p^{n}-1)}$ acts through $v_{n}$ [25, Remark 2.1.2].

To relate particular forms of $BP\langle n\rangle$ to Lubin-Tate spectra, we
use the spherical Witt vectors constructed by Lurie [37, Example 5.2.7]. For a
given discrete perfect $\mathbb{F}_{p}$-algebra $B_{0}$, this provides an
$E_{\infty}$-ring $\mathbb{S}_{W(B_{0})}$ that is flat over $\mathbb{S}$ in
the sense of [36, Defnition 7.2.2.10]. Therefore, it follows by [36,
Proposition 7.2.2.13] and [38, Proposition 2.7] that

(8.1) $\pi_{n}(\mathbb{S}_{W(B_{0})}\wedge F)\cong W(B_{0})\otimes\pi_{n}F.$

for every spectrum $F$. We would like to thank Jeremy Hahn for showing us the
proof of the following proposition.

######  Proposition 8.2.

Fix an $E_{3}$ $MU$-algebra form of $BP\langle n\rangle$. Then
$\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle$ satisfies the redshift
conjecture for every discrete perfect $\mathbb{F}_{p}$-algebra $B_{0}$.

###### Proof.

There is an equivalence of $E_{2}$-algebras in $S^{1}$-equivariant spectra

$\mathbb{S}_{W(B_{0})}\wedge\operatorname{\textup{THH}}^{MU}(BP\langle
n\rangle)\simeq\operatorname{\textup{THH}}^{\mathbb{S}_{W(B_{0})}\wedge
MU}(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle).$

Therefore, it follows by (8.1) that we have

(8.3) $\operatorname{\textup{THH}}^{\mathbb{S}_{W(B_{0})}\wedge
MU}_{*}(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle)\cong
W(B_{0})\otimes\operatorname{\textup{THH}}^{MU}_{*}(BP\langle n\rangle)$

and this is a polynomial algebra over $W(B_{0})[v_{1},...,v_{n}]$ due to [25,
Theorem 2.5.4] where one of the generators is denoted by $\sigma^{2}v_{n+1}$ .
The rest of the argument follows as in the proofs of [25, Theorems 2.5.4 and
5.0.1, Corollary 5.0.2] by considering

(8.4) $\pi_{*}\big{(}\operatorname{\textup{THH}}^{\mathbb{S}_{W(B_{0})}\wedge
MU}(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle)^{hS^{1}}\big{)}$

instead of $\pi_{*}\big{(}\operatorname{\textup{THH}}^{MU}(BP\langle
n\rangle)^{hS^{1}}\big{)}$.

Namely, we obtain from [25, Theorem 2.5.4] that (8.3) is concentrated in even
degrees. Therefore, the corresponding homotopy fixed point spectral sequence
degenerates on the second page providing the
$W(B_{0})[v_{1},...,v_{n}]$-algebra (8.4) as

$W(B_{0})\otimes\operatorname{\textup{THH}}^{MU}_{*}(BP\langle
n\rangle)\llbracket t\rrbracket$

since (8.3) is polynomial. Using the map from
$\pi_{*}\big{(}\operatorname{\textup{THH}}^{MU}(BP\langle
n\rangle)^{hS^{1}}\big{)}$ to (8.4), we deduce from [25, Theorem 5.0.1] that
$v_{n+1}$ in (8.4) is represented by the class $t\sigma^{2}v_{n+1}$.

Considering the action of $v_{0},...v_{n+1}$ on (8.4) described above, one
observes that

$L_{T(n+1)}\operatorname{\textup{THH}}^{\mathbb{S}_{W(B_{0})}\wedge
MU}(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle)^{hS^{1}}\not\simeq*.$

Using this and the $E_{2}$-map

$L_{T(n+1)}K(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle)\to
L_{T(n+1)}\operatorname{\textup{THH}}^{\mathbb{S}_{W(B_{0})}\wedge
MU}(\mathbb{S}_{W(B_{0})}\wedge BP\langle n\rangle)^{hS^{1}},$

we deduce that $L_{T(n+1)}K(\mathbb{S}_{W(B_{0})}\wedge BP\langle
n\rangle)\not\simeq*$ as desired. ∎

######  Construction 8.5.

Fix an $E_{3}$ $MU[\sigma_{2(p^{n}-1)}]$-algebra form of $BP\langle n\rangle$
where $\sigma_{2(p^{n}-1)}$ acts through $v_{n}$ and let $k$ be a perfect
field of characteristic $p$. Recall that in this situation, degree
$p^{n}-1$-root adjunction to $v_{n}$ provides a $p^{n}-1$-graded $E_{3}$
$MU[\sigma_{2}]$-algebra, see Remark 4.11. We consider the $E_{3}$
$MU$-algebra:

$E:=\big{(}L_{K(n)}(\mathbb{S}_{W(k)}\wedge BP\langle
n\rangle)\big{)}(\sqrt[p^{n}-1]{v_{n}}).$

It follows by [22, Theorem 1.5.4] and [21, Theorem 1.9] that

$\pi_{*}E\cong W(k)[\lvert u_{1},...,u_{n-1}\rvert][u^{{\pm 1}}]$

where $\lvert u_{i}\rvert=0$ and $\lvert u\rvert=-2$. Furthermore, the
resulting $E_{3}$-map $MU\to BP\langle n\rangle\to E$ provides a formal group
law over $\pi_{*}E$.

For a given perfect $\mathbb{F}_{p}$-algebra $B_{0}$ and a height $n$ formal
group law $\Gamma$ over $B_{0}$, we let $E_{(B_{0},\Gamma)}$ denote the
corresponding Lubin-Tate spectrum. By Lurie’s generalization [37, Section 5]
of the Goerss-Hopkins-Miller theorem [44, 16], $E_{(B_{0},\Gamma)}$ is an
$E_{\infty}$-ring.

######  Proposition 8.6.

In the setting of Construction 8.5, $E$ is equivalent to $E_{(k,\Gamma)}$ as
an $E_{1}$-ring for some height $n$ formal group law $\Gamma$ over $k$.

###### Proof.

By construction, the map $\pi_{*}MU_{(p)}\to\pi_{*}E$ carries $v_{i}$ to
$u_{i}u^{-(p^{i}-1)}$ for $0<i<n$ and $v_{n}$ to $u^{-(p^{n}-1)}$. Therefore,
the corresponding formal group law on $\pi_{0}E$ is the universal deformation
of the resulting height $n$ formal group law $\Gamma$ over $k$. This follows
by [34, Theorem 5 in Lecture 21]. Alternatively, one can directly check the
conditions given in [32, Proposition 1.1]. It follows from Hopkins-Miller
theorem that there is an equivalence of $E_{1}$-rings $E\simeq E_{(k,\Gamma)}$
[44, Theorem 7.1].

∎

Burklund, Hahn, Levy and Schlank are going to use the following example in
their construction of a counterexample to the telescope conjecture.

######  Example 8.7.

Let $k$ be a perfect algebraic extension of $\mathbb{F}_{p}$. We know from
[43, Corollary 4.31] that the $E_{1}$-algebra structure on $E_{(k,\Gamma)}$
lifts to a unique $E_{d}$-algebra structure for every $1\leq d\leq\infty$.
Since $E$ in Construction 8.5 is an $E_{3}$-ring, it follows from Proposition
8.6 that there is an $E_{3}$-equivalence

$E\simeq E_{(k,\Gamma)}$

for $\Gamma$ as in Proposition 8.6. Through this equivalence, we may equip
$E_{(k,\Gamma)}$ with the structure of an $E_{3}$ $MU$-algebra. In particular,
we obtain a map of $E_{3}$ $MU$-algebras

$BP\langle n\rangle\to E_{(k,\Gamma)}.$

Furthermore, $E_{(k,\Gamma)}$ is a $\mathbb{Z}/(p^{n}-1)$-graded $E_{3}$
$MU[\sigma_{2}]$-algebra where the weight $1$ class $\sigma_{2}$ acts through
$u^{-1}$.

######  Theorem 8.8.

In the setting of Construction 8.5 and Proposition 8.6, the canonical map

$L_{T(n+1)}K(\mathbb{S}_{W(k)}\wedge BP\langle n\rangle)\to
L_{T(n+1)}K(E_{(k,\Gamma)})$

is the inclusion of a non-trivial wedge summand.

###### Proof.

It follows by Corollary 5.11 and Proposition 8.6 that

$L_{T(n+1)}K\big{(}L_{K(n)}(\mathbb{S}_{W(k)}\wedge BP\langle
n\rangle)\big{)}\to L_{T(n+1)}K(E_{(k,\Gamma)})$

is the inclusion of a wedge summand. Since

$\mathbb{S}_{W(k)}\wedge BP\langle n\rangle\to
L_{K(n)}(\mathbb{S}_{W(k)}\wedge BP\langle n\rangle)$

is a $T(n)\vee T(n+1)$-equivalence, the result follows by [31, Purity
Theorem]. ∎

We finally prove the following theorem of Yuan.

######  Theorem 8.9 ([52]).

For every perfect $\mathbb{F}_{p}$-algebra $B_{0}$ and height $n$ formal group
law $\Gamma$ over $B_{0}$, the Lubin-Tate spectrum $E_{(B_{0},\Gamma)}$
satisfies the redshift conjecture.

###### Proof.

There is an $E_{\infty}$-map $E_{(B_{0},\Gamma)}\to E_{(k,\Gamma^{\prime})}$
for some algebraically closed field $k$ of characteristic $p$ and
$\Gamma^{\prime}$ is the corresponding height $n$ formal group law on $k$.
Since there is an induced $E_{\infty}$-map $K(E_{(B_{0},\Gamma)})\to
K(E_{(k,\Gamma^{\prime})})$, it suffices to prove the redshift conjecture for
$E_{(k,\Gamma^{\prime})}$.

Since $k$ is algebraically closed, there is a unique formal group law of
height $n$ over $k$ [29, Theorem IV]. Using Proposition 8.6, we deduce that
$E_{(k,\Gamma^{\prime})}\simeq E$ for $E$ as in Construction 8.5. Combining
Proposition 8.2 and Theorem 8.8 we obtain that $E_{(k,\Gamma^{\prime})}$
satisfies the redshift conjecture as desired. ∎

We remark that it should be possible to prove the redshift conjecture for all
$E_{1}$ $MU$-algebra forms of $BP\langle n\rangle$ by constructing maps
$BP\langle n\rangle\to E_{(k,\Gamma)}$ through root adjunction.

##  9\. Algebraic $K$-theory and THH of Morava $E$-theories

In this section, we work with a particular form of Lubin-Tate spectra. This is
the Morava $E$-theory spectrum $E_{n}$ and $E_{n}$ is central in the Ausoni-
Rognes program for the computation of $K(\mathbb{S})$. When we say Morava
$E$-theory $E_{n}$, we mean the Lubin-Tate spectrum corresponding to the
height $n$ Honda formal group. This formal group is characterized by its
$p$-series

$[p]_{n}(x)=x^{p^{n}},$

and admits a canonical form over $\mathbb{F}_{p^{n}}$, in the sense that all
of its endomorphisms are defined over this field. In this section, we prove a
splitting result for the algebraic $K$-theory of the Morava $E$-theory $E_{n}$
and the corresponding two periodic Morava $K$-theory. Furthermore, we show
that the THH of $E_{n}$ may be obtain from the THH of the $K(n)$-localized
Johnson-Wilson spectrum through base change.

###  9.1. An identification of Morava $E$-theory

Here, we provide an alternate description of $E_{n}$ in terms of its fixed
points and spherical Witt vectors. We have

$\pi_{*}E_{n}\cong W(\mathbb{F}_{p^{n}})[\lvert
u_{1},\dots,u_{n-1}\rvert][u^{\pm 1}]$

where $\lvert u_{i}\rvert=0$ and $\lvert u\rvert=-2$.

######  Proposition 9.1.

The map

$\pi_{*}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\cong
W(\mathbb{F}_{p^{n}})\otimes_{\mathbb{Z}_{p}}\pi_{*}\mathbb{S}_{p}\to\pi_{*}E_{n}$

obtained via the map $\pi_{*}\mathbb{S}_{p}\to\pi_{*}E_{n}$ and the canonical
$W(\mathbb{F}_{p^{n}})$-module structure on $\pi_{*}E_{n}$ lifts to a map of
$E_{\infty}$ $\mathbb{S}_{p}$-algebras

$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\to E_{n}.$

###### Proof.

This is a consequence of Lurie’s theory of thickenings of relatively perfect
morphisms [37, Section 5.2]. Indeed,
$\mathbb{S}_{p}\to\mathbb{S}_{W(\mathbb{F}_{p^{n}})}$ is an
$\mathbb{S}_{p}$-thickening of $H\mathbb{F}_{p}\to H\mathbb{F}_{p^{n}}$ in the
sense of [37, Definition 5.2.1], see [37, Example 5.2.7].

In particular, this implies that the space of $E_{\infty}$
$\mathbb{S}_{p}$-algebra maps from $\mathbb{S}_{W(\mathbb{F}_{p^{n}})}$ to the
connective cover $cE_{n}$ of $E_{n}$ is given by the set of
$\mathbb{F}_{p}$-algebra maps

(9.2)
$\hom_{\mathbb{F}_{p}\textup{-}\mathcal{A}\textup{lg}}(\mathbb{F}_{p^{n}},\mathbb{F}_{p^{n}}[\lvert
u_{1},\dots,u_{n-1}\rvert])$

where this correspondence is given by the functor $\pi_{0}(-)/p$.

Let $f\colon\thinspace\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\to cE_{n}$ be the map
of $E_{\infty}$ $\mathbb{S}_{p}$-algebras corresponding to the canonical
$\mathbb{F}_{p}$-algebra map in (9.2); in particular, $\pi_{0}(f)/p$ is the
canonical map in (9.2). We first show that $\pi_{0}f$ is given by the
canonical inclusion

$W(\mathbb{F}_{p^{n}})\to W(\mathbb{F}_{p^{n}})[\lvert
u_{1},\dots,u_{n-1}\rvert].$

Since $W(\mathbb{F}_{p})$ is the ring of integers of the unique unramified
extension $\mathbb{Q}_{p}[\mu_{p^{n}-1}]$ of $\mathbb{Q}_{p}$ of degree $d$,
$W(\mathbb{F}_{p^{n}})$ is generated as a $\mathbb{Z}_{p}$-algebra by a
primitive $p^{n}-1$ root of the unit. Since the roots of the unit of
$\pi_{0}cE_{n}$ are all in the image of the canonical inclusion
$W(\mathbb{F}_{p^{n}})\to W(\mathbb{F}_{p^{n}})[\lvert
u_{1},\dots,u_{n-1}\rvert]$, one observes that the map $\pi_{0}f$ has to
factor through the canonical inclusion $W(\mathbb{F}_{p^{n}})\to
W(\mathbb{F}_{p^{n}})[\lvert u_{1},\dots,u_{n-1}\rvert]$. Furthermore, there
is a unique ring map $W(\mathbb{F}_{p^{n}})\to W(\mathbb{F}_{p^{n}})$ that
lifts the identity map on $\mathbb{F}_{p^{n}}$. This shows that $\pi_{0}f$ is
given by the canonical inclusion $W(\mathbb{F}_{p^{n}})\to
W(\mathbb{F}_{p^{n}})[\lvert u_{1},\dots,u_{n-1}\rvert]$. Since $\pi_{*}f$ is
a map of $\pi_{*}\mathbb{S}_{p}$-modules, it follows that the composition of
$f$ with the map $cE_{n}\to E_{n}$ provides the map claimed in the
proposition.

∎

Let $Gal$ denote the Galois group
$\text{Gal}(\mathbb{F}_{p^{n}},\mathbb{F}_{p})$. Due to Goerss-Hopkins-Miller
theorem, there is an action of $Gal$ on the $E_{\infty}$-algebra $E_{n}$ for
which

$\pi_{*}E_{n}^{hGal}\cong\mathbb{Z}_{p}[\lvert
u_{1}\dots,u_{n-1}\rvert][u^{\pm 1}]$

where the degrees of the generators are as in $\pi_{*}E_{n}$.

######  Proposition 9.3.

There is an equivalence of $E_{\infty}$-$\mathbb{S}_{p}$-algebras:

$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}E_{n}^{hGal}\simeq
E_{n}.$

###### Proof.

This equivalence is given by the following composite map of $E_{\infty}$
$\mathbb{S}_{p}$-algebras

(9.4)
$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}E_{n}^{hGal}\to
E_{n}\wedge_{\mathbb{S}_{p}}E_{n}\to E_{n}$

where the first map is induced by the map provided by Proposition 9.1 and the
second map is given by the multiplication map of $E_{n}$. Due to the flatness
of $\mathbb{S}_{W(\mathbb{F}_{p^{n}})}$, this map induces the canonical map

(9.5) $W(\mathbb{F}_{p^{n}})\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}[\lvert
u_{1},\dots,u_{n-1}\rvert][u^{\pm 1}]\to W(\mathbb{F}_{p^{n}})[\lvert
u_{1},\dots,u_{n-1}\rvert][u^{\pm 1}].$

at the level of homotopy groups [36, 7.2.2.13]. Since $W(\mathbb{F}_{p^{n}})$
is a free $\mathbb{Z}_{p}$-module of finite rank, the functor
$W(\mathbb{F}_{p^{n}})\otimes_{\mathbb{Z}_{p}}-$ is given by taking a $n$-fold
product of $-$. In particular, the functor
$W(\mathbb{F}_{p^{n}})\otimes_{\mathbb{Z}_{p}}-$ commutes with completions.
This shows that (9.5) is an isomorphism as desired. ∎

###  9.2. Algebraic $K$-theory of Morava $E$-theories

There is a finite subgroup $\mathbb{F}_{p^{n}}^{\times}$ of the Morava
stabilizer group such that $K=\mathbb{F}_{p^{n}}^{\times}\rtimes Gal$ acts on
the $E_{\infty}$-algebra $E_{n}$. Furthermore,

$E_{n}^{hK}\simeq\widehat{E(n)}$

where $\widehat{E(n)}$ denotes the $K(n)$-localization of the Johnson-Wilson
spectrum $E(n)$, see [45, Section 5.4.7]. We have

$\pi_{*}E(n)\cong\mathbb{Z}_{(p)}[v_{1},\dots,v_{n-1}][v_{n}^{\pm 1}]\textup{\
\ and \
}\pi_{*}\widehat{E(n)}\cong\mathbb{Z}_{(p)}[v_{1},\dots,v_{n-1}][v_{n}^{\pm
1}]_{I}^{\land}$

where $I$ denotes the ideal $(p,v_{1},\dots,v_{n-1})$. Since $\widehat{E(n)}$
is given by $E_{n}^{hK}$, there is a $K$-equivariant map of
$E_{\infty}$-algebras $\widehat{E(n)}\to E_{n}$. In particular, this provides
a map

$\widehat{E(n)}\to E_{n}^{hGal}$

of $E_{\infty}$-algebras. This map carries $v_{n}$ to $u^{-(p^{n}-1)}$ and
$v_{i}$ to $u_{i}u^{-(p^{i}-1)}$ for $0<i<n$.

For the following, we fix an $E_{2}$-map
$\mathbb{S}[\sigma_{2(p^{n}-1)}]\to\widehat{E(n)}$ to adjoin roots. Recall
from Remark 4.8 that in this situation,
$\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})$ is an $\widehat{E(n)}$-algebra.

######  Theorem 9.6.

There are equivalences of $E_{1}$ $\widehat{E(n)}$-algebras:

$\displaystyle E_{n}^{hGal}\simeq$ $\displaystyle\
\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})$ $\displaystyle E_{n}\simeq$
$\displaystyle\
\mathbb{S}_{W(\mathbb{F}_{q})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})$

where the class $u^{-1}$ corresponds to $\sqrt[p^{n}-1]{v_{n}}$ at the level
of homotopy groups for both of these equivalences.

In particular, $E_{n}^{hGal}$ and $E_{n}$ are $p^{n}-1$-graded $E_{1}$
$\widehat{E(n)}$-algebras with

$(E_{n}^{hGal})_{i}\simeq\Sigma^{2i}\widehat{E(n)}$

and

$(E_{n})_{i}\simeq\Sigma^{2i}\mathbb{S}_{W(\mathbb{F}_{q})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}$

for every $0\leq i<p^{n}-1$.

###### Proof.

By inspection, one observes that

$\pi_{*}(\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}}))\cong\pi_{*}E_{n}^{hGal},$

see [45, 5.4.9]. Furthermore, the map of rings,

$\pi_{*}\widehat{E(n)}\to\pi_{*}(\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}}))\cong(\pi_{*}\widehat{E(n)})[z]/(z^{p^{n}-1}-v_{n})$

is a map of étale rings as $v_{n}$ and $p^{n}-1$ are invertible in
$\pi_{*}\widehat{E(n)}$. Through [23, Theorem 1.10], we obtain the first
equivalence in the theorem. The second equivalence follows by the first
equivalence and Proposition 9.3. The statement on graded ring structures
follows by the fact that root adjunction results in $m$-graded ring spectra,
see Construction 4.6. ∎

We are ready to prove our result on the $K$-theory of Morava $E$-theories. For
this, we use the following composite map

(9.7)
$E(n)\to\widehat{E(n)}\to\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})\xrightarrow{\simeq}E_{n}^{hGal}$

of $E_{1}$-rings where the last map above is given by Theorem 9.6. Using
Proposition 9.3, we obtain the following composite:

(9.8) $\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge
E(n)\to\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge\widehat{E(n)}\to\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge
E_{n}^{hGal}\to\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}E_{n}^{hGal}\xrightarrow{\simeq}E_{n}.$

######  Theorem 9.9.

The maps

$\displaystyle K(E(n))\to$ $\displaystyle\ K(E_{n}^{hGal})$ $\displaystyle
K(\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge E(n))\to$ $\displaystyle\ K(E_{n})$

induced by those above are inclusions of wedge summands after
$T(n+1)$-localization.

###### Proof.

The first map in (9.7) is a $T(n)\vee T(n+1)$-equivalence and hence induces a
$T(n+1)$-equivalence in algebraic $K$-theory [31, Purity Theorem]. Therefore,
the first equivalence in the theorem follows by applying Corollary 5.11 to
$\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})$.

Similarly, the first and the third maps in (9.8) induce $T(n+1)$-equivalences
in algebraic $K$-theory. The second equivalence in the theorem follows by
observing that there is an equivalence of $E_{1}$-algebras:

$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge\big{(}\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}})\big{)}\simeq\big{(}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge\widehat{E(n)}\big{)}(\sqrt[p^{n}-1]{v_{n}})$

and then applying Corollary 5.11. ∎

###  9.3. Two-periodic Morava $K$-theories

We obtain analogous results for two-periodic Morava K-theories. Taking a
quotient with respect to a regular sequence in
$\pi_{*}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}$,
one obtains an $\widehat{E(n)}$-algebra $K(n)$ [30, 3, 24]. Here, $K(n)$ is
the Morava $K$-theory spectrum with coefficients in $\mathbb{F}_{p^{n}}$. We
have

$\pi_{*}K(n)\cong\mathbb{F}_{p^{n}}[v_{n}^{{\pm 1}}].$

Using the $\widehat{E(n)}$-algebra structure on $K(n)$, we adjoin roots and
define the two periodic Morava $K$-theory as follows

$K_{n}:=K(n)(\sqrt[p^{n}-1]{v_{n}}).$

In this case,

$\pi_{*}K_{n}\cong\mathbb{F}_{p^{n}}[u^{{\pm 1}}]$

where $\lvert u\rvert=-2$. Together with Theorem 9.6, this provides a
commuting diagram of $E_{1}$-rings

${\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\wedge_{\mathbb{S}_{p}}\widehat{E(n)}}$${E_{n}}$${K(n)}$${K_{n}}$

which justifies our definition of $K_{n}$. In particular, $K_{n}$ is a
$p^{n}-1$-graded $E_{1}$-ring in a non-trivial way and we obtain the following
from Corollary 5.11.

######  Theorem 9.10.

The following map

$K(K(n))\to K(K_{n})$

is the inclusion of a wedge summand after $T(i)$-localization for every $i\geq
2$.

######  Corollary 9.11.

If $K(n)$ satisfies the redshift conjecture, then so does $K_{n}$.

The $V(1)$-homotopy of $K(k(1))$ is computed by Ausoni and Rognes in [5] for
$p>3$. In particular, their computation shows that $K(1)$ satisfies the
redshift conjecture. We obtain the following.

######  Corollary 9.12.

The two periodic Morava $K$-theory $K_{1}$ of height one satisfies the
redshift conjecture for $p>3$.

###  9.4. THH descent for Morava $E$-theories

Theorem 6.28 identifies THH of various periodic ring spectra with their
logarithmic THH. For instance, the Morava $E$-theory spectrum $E_{n}$ is
periodic with a unit $u$ in degree $-2$. Since $E_{n}/(u^{-1})\simeq 0$, the
canonical map

$\operatorname{\textup{THH}}(E_{n})\xrightarrow{\simeq}\operatorname{\textup{THH}}(E_{n}\mid
u^{-1})$

is an equivalence. Using this, together with our result on logarithmic THH-
étaleness of root adjunction, we show that
$\operatorname{\textup{THH}}(E_{n})$ may be obtained from
$\operatorname{\textup{THH}}(\widehat{E(n)})$ via base-change up to
$p$-completion. Such base-change formulas and their relationship with Galois
descent problems for THH were studied by Mathew in [39].

######  Theorem 9.13.

The canonical map:

$\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}E_{n}^{hGal}\xrightarrow{\simeq}\operatorname{\textup{THH}}(E_{n}^{hGal}),$

is an equivalence.

###### Proof.

Recall from Theorem 9.6 that there is an equivalence of
$\widehat{E(n)}$-algebras

$E_{n}^{hGal}\simeq\widehat{E(n)}(\sqrt[p^{n}-1]{v_{n}}).$

Therefore, it follows by Construction 4.6 that

(9.14)
$\begin{split}\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}E_{n}^{hGal}&\simeq\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}\widehat{E(n)}\wedge_{\mathbb{S}_{(p)}[\sigma_{2(p^{n}-1)}]}\mathbb{S}_{(p)}[\sigma_{2}]\\\
&\simeq\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\mathbb{S}_{(p)}[\sigma_{2(p^{n}-1)}]}\mathbb{S}_{(p)}[\sigma_{2}].\end{split}$

Since $v_{n}$ is a unit in $\widehat{E(n)}$ and $u^{-1}$ is a unit in
$E_{n}^{hGal}$, Theorem 6.28 provides the equivalences:

$\operatorname{\textup{THH}}(\widehat{E(n)})\xrightarrow{\simeq}\operatorname{\textup{THH}}(\widehat{E(n)}\mid
v_{n})\textup{\ \ and \
}\operatorname{\textup{THH}}(E_{n}^{hGal})\xrightarrow{\simeq}\operatorname{\textup{THH}}(E_{n}^{hGal}\mid
u^{-1}).$

Using these equivalences together with Theorem 6.23, we obtain that the
following canonical map is an equivalence.

$\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\mathbb{S}_{(p)}[\sigma_{2(p^{n}-1)}]}\mathbb{S}_{(p)}[\sigma_{2}]\xrightarrow{\simeq}\operatorname{\textup{THH}}(E_{n}^{hGal})$

This, together with (9.14), provides the desired result. ∎

######  Theorem 9.15.

The canonical map:

$\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}E_{n}\xrightarrow{\simeq_{p}}\operatorname{\textup{THH}}(E_{n}),$

is an equivalence after $p$-completion.

###### Proof.

The first equivalence below follows by Proposition 9.3. and the second follows
by Theorem 9.13.

$\begin{split}\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}E_{n}&\simeq\operatorname{\textup{THH}}(\widehat{E(n)})\wedge_{\widehat{E(n)}}(E_{n}^{hGal}\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})})\\\
&\simeq\operatorname{\textup{THH}}(E_{n}^{hGal})\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\end{split}$

Therefore, it is sufficient to show that the canonical map

$\operatorname{\textup{THH}}(E_{n}^{hGal})\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\to\operatorname{\textup{THH}}(E_{n}^{hGal}\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})})$

is an equivalence after $p$-completion. This follows by the following
canonical diagram of $\mathbb{S}/p$-equivalences.

${\operatorname{\textup{THH}}(E_{n}^{hGal})\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}}$${\operatorname{\textup{THH}}(E_{n}^{hGal}\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})})}$${\operatorname{\textup{THH}}^{\mathbb{S}_{p}}(E_{n}^{hGal})\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})}}$${\operatorname{\textup{THH}}^{\mathbb{S}_{p}}(E_{n}^{hGal}\wedge_{\mathbb{S}_{p}}\mathbb{S}_{W(\mathbb{F}_{p^{n}})})}$${\operatorname{\textup{THH}}^{\mathbb{S}_{p}}(E_{n}^{hGal})\wedge_{\mathbb{S}_{p}}\operatorname{\textup{THH}}^{\mathbb{S}_{p}}(\mathbb{S}_{W(\mathbb{F}_{p^{n}})})}$$\scriptstyle{\simeq_{p}}$$\scriptstyle{\simeq_{p}}$$\scriptstyle{\simeq_{p}}$$\scriptstyle{\simeq}$

The right hand vertical map and the upper left vertical map are
$\mathbb{S}/p$-equivalences due to [38, Lemma 5.20]. The fact that the lower
left vertical map is an $\mathbb{S}/p$-equivalence follows by [38, proof of
Lemma 5.20] and the fact that the composite
$\mathbb{S}_{W(\mathbb{F}_{p^{n}})}\to\operatorname{\textup{THH}}(\mathbb{S}_{W(\mathbb{F}_{p^{n}})})\to\mathbb{S}_{W(\mathbb{F}_{p^{n}})}$
is an equivalence. This shows that the upper horizontal map is an
$\mathbb{S}/p$-equivalence proving the theorem. ∎

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# First 3D reconstruction of a blast furnace using muography

A. Cohu  A. Chevalier  O. Nechyporuk  A. Franzen  J. Sauerwald  J-C. Ianigro
and J. Marteau

###### Abstract

The blast furnace (BF) is the fundamental tool used in the iron manufacture.
Due to the difficulty of accessing direct measurements of the inner phenomena,
we determined the density distribution of its internal volume in order to
improve its productivity using muography. This is an imaging technique based
on the differential absorption of a flux of incident particles, muons, by the
target under study, similar to clinical X-ray imaging. Muons are elementary
particles that have the property of passing through dense materials, up to
hundreds of meters away. Their relative absorption and deviation allows the
generation of density distribution images of an object by tracking the number
of muons received by a detector, before and after passing through a structure.
The incident direction of the detected muons is reconstructed by means of a
detector composed of 3 scintillator panels that we moved on 3 positions around
the BF. With this technique, we obtained the first 3D image of the internal
structure of a BF using a Markov Chain Monte Carlo (MCMC) inverse problem
solving algorithm on muon flux data. We were also able to perform a density
monitoring of the BF and some of its operating parameters. We distinguished
the position and shape of the cohesive zone, a key element in the productivity
of a furnace, validating this innovative measurement concept in the
application to a BF and opening the field to a series of future experiments to
gain both spatial and temporal resolution.

##  1 Introduction

We seek to determine the density distribution of matter inside a blast furnace
in order to visualize the cohesive zone and to carry out a dynamic monitoring
of the various phases present in the blast furnace. To achieve this, muography
is applied to make a dynamic image during the operation of a blast furnace of
ArcelorMittal in Bremen, Germany. Muography measures the absorption or
deflection of cosmic muons as they pass through dense materials. Muons are
elementary particles which have the property to pass, in a straight line at
first order, up to several kilometers of standard rocks, and whose relative
absorption allows to generate images by contrast densitometry, like a standard
clinical radiography. The acquired muon data allows to follow the density as a
function of time during the operating cycles of the blast furnace. In a second
step, the acquisition of 2D images and then the 3D reconstruction is
accomplished by the data inversion from several measurement points. The final
objective is to understand the topological characteristics and the formation
rate of the cohesive zone and the influence of certain loading parameters.

The article is organized in four main parts. The first part explains the
theorical points and algorithms used in tomography reconstructions. The second
part presents the simulation parameters of a blast furnace and explains how we
built the 3D image. We have completed our analysis by monitoring the activity
of the blast furnace as function of different environmental parameters such as
atmospheric pressure and temperature. The third part presents the results of
real muons data inversion to visualize the different density zones in the
blast furnace. Finally, we report the conclusions and perspectives of this
study.

##  2 Tomography reconstruction theory

###  2.1 Generalities and issues

Absorption muography measures the cosmic ray flux deficit in the direction of
observation and determines the integrated density of a structure. Muons are
cosmic rays able to cross several meters of rocks losing energy. The minimum
amount of energy that muons must need to penetrate the structure must be of a
value higher than the one lost inside the object, so that the detector can
follow the outgoing muons. The detector position must then be adjusted to
optimize the spatial resolution during field measurements. Muon tomography is
limited to the study of a portion of the object only, because of the limited
angular aperture of the detector. In order to take these different points into
consideration, we use the inverse and direct problems jointly.

The direct problem consists, in our case, in predicting the expected muon flux
at the exit of an object. It is necessary to use materials knowledge and
physical properties of the object’s constituents and then to use a density
distribution as precise as possible. The parameters sought during the
inversion use the direct problem to estimate the expected measurements. The
measured information is retrieved from a given distribution of $p$ parameters
in the studied structure. Moreover, the flux attenuation is estimated from a
known law based on the contrasting distribution of zones (of different density
for example).

With the trackers the direction of the muons is reconstructed in order to
observe the properties of the medium on a precise observation axis (see
Lesparre et al.[13]). The flux that arrives on the detector after having
crossed the object and the theoretical flux that would reach the detector in
the absence of matter are compared. The contrast between thoses two quantities
gives directly access to the opacity of the matter (in meter equivalent water
(mwe)), defined as the integral of the density along the trajectory of the
muon from its entry point to its exit point. Moreover, the only observable are
the particles directions and energy deposits, since the detectors usually do
not give direct information on the particles total energy. In order to solve
the data inversion, the measured flux is coupled to a theoretical flux model
and to a flux loss model in matter.

###  2.2 Inverse problem

The reconstruction of a muography image is achieved by solving an inverse
problem. The goal is to recover the distribution of properties of the medium
(3D density) from measurements of muon rates and their directions. An inverse
problem is a situation in which one tries to determine the parameters of a
model $p$ (here the 3D density of the environnement) from experimental
measurements $m$ (muon rates) such that $m=f(p)$ where $f$ contains the open
muon flux (calibration) and the law governing the absorption of muons in
matter. While a single parameter estimate can be easily obtained by least
squares fitting, the use of Monte Carlo methods allows the maintenance of the
stochasticity of the model during the estimation of these parameters. In order
to improve the reliability of the results, it is good practice to add some
informations about the objet to study, other than data, that we call a priori
information [1]. They allow to constrain the solutions space and improve the
accuracy of the statistical answer.

We use the Metropolis-Hastings algorithm which is a particular class of Monte
Carlo methods using Markov chains. It works like a geometrically biased random
walk with a data based selection at each throw [8]. Each new proposal/model
can be accepted or rejected if the likelihood of the model (regarding the data
and the physics of the problem) is greater or smaller than the likelihood of
the previous model. Hence, unlike more general Monte Carlo methods where the
sampled values are statistically independent, in the Metropolis-Hastings
algorithm they are statistically auto-correlated. This auto-correlation is
minimized by adding a threshold number of changes that must be accepted
between the recording of two consecutive samples.

We need models whose density values are continuous over finite element
discretized volumes. Here a model stands for any given set of values
representing a physical system. The engine that generates the 3D density
models, linked by Markov chain, is made to design density sets per class
forming spatially contiguous voxel sets which share the same density. The
inversion algorithm is a 3D adaptation of Mosegaard et al.[11] work, where
details can be found in Chevalier et al.[7]. The method makes selections of
the models according to the evolution of the distance $D$ :

$D=\frac{F_{t}-F_{m}}{\sigma_{m}}$ (2.1)

with $F_{t}$ the theoretical flux and $F_{m}$ the measured flux. $D$ is a
metric of the distance between the data and the simulation, expanded or
compressed by the degree of uncertainty on the measurement $\sigma_{m}$. This
distance $D$ is recalculated each time the density changes. A new model is
accepted as a solution if the mean difference between the reconstructed signal
and the data (evaluated by the mean square deviation) is lower than the mean
noise estimated during the measurement. If the model is selected, we save its
likelihood. This model is then slightly perturbed again by changing some voxel
density values and we recalculate the new likelihood. The total likelihood of
a model, $L(m)$, can be expressed as a product of partial likelihoods, one for
each data type.

Our inversion algorithm is able to couple information from several detectors
at the same time. Indeed, each detector measures a flux andtravel length
(defined as the thickness of material seen by the detector in its acquisition
configuration) per viewing axis. By taking a set of density distributed in the
voxels, we have the opacity seen by the viewing axis. Finally, the number of
registered models can be modified and it depends on the requirements in terms
of accuracy and available computing time.

##  3 Application to a blast furnace

###  3.1 Acquisition configurations

In order to obtain a 3D image of the blast furnace, we carried out three
muography acquisitions around the blast furnace, performed between the end of
July 2021 and the end of March 2022 with the 3 planes detector (specifications
of the runs are described in table 1). In this way we intersect a maximum
volume of it, common to the fields of view of the three detectors. On figure
1, the detector virtual lines of sigth at each position are shown. They allow
visualization of the field of view of each location and the common areas that
are observed.

Figure 1: Detector virtual lines of sight at each position with different colors : position/run 1 (in red), position 2 (in blue), position 3 (in green).  Table 1: Specifications of the 3 runs shown in figure 1(Ze=zenith, Az=azimuth angles).  Run | Ze (°) | Az (°) | X (m) | Y (m) | Z (m) | Dates
---|---|---|---|---|---|---
1 | 45 | 51 | -7.51 | -7.04 | 11.35 | 07/29/21 - 20/09/21
2 | 60 | 144.33 | -13.49 | 9.52 | 14.55 |  | 10/08/21- 10/21/21
---
& 12/21/21 - 01/24/22
3 | 60 | 270 | 15.14 | 0.24 | 14.55 | 02/03/22 - 03/31/22

###  3.2 External and internal parameters effects on muon flux

By counting the muons rate, in a given direction as a function of time for
each of the positions shown on figure 1, we observe temporal variations. The
rate of high-energy cosmic ray muons as measured underground is shown to be
strongly correlated with upper-air temperatures during short-term atmospheric
phenomena [3, 2]and with pressure [5]. We have evaluated the effects of
atmospheric parameters and those of the target operation (coke rate estimate
at the cohesive zone and the value of the blast pressure, defined above). Our
goal is to substract these different effects from the measured rate, before
realizing the data inversion, to obtain a density distribution 3D image in the
blast furnace.

We chose to compute the pressure/flux correlations for each position
separately because the detectors are not sensitive to the same opacity from
one location to another. By performing a linear fit between the flux and
ambient pressure data, we obtain the barometric coefficients $\beta_{p}$
($\mathrm{h}\mathrm{P}\mathrm{a}^{-1}$), such as
$\dfrac{(\phi-\phi_{0})}{/}{\phi_{0}}=\beta_{p}\times(P-P_{0})$ (see Jourde et
al.[5]), their error as well as the correlation coefficient of the fit in the
table 2. The correlation values between pressure and flux vary with time
according to the table 2 and the sensitivity of the muon flux to the pressure
variation appears more important in high pressure episodes. Moreover, the muon
fluxes do not seem to be sensitive to the temperature variations of the upper
atmosphere over the studied periods and they are thus more affected by the
pressure variations. This result is consistent with the opacity values
encountered, at most 50 mwe (lower than those of Tramontini et al. [2] $\sim$
700 mwe and close to Acernese [4] open sky experiment) which do not allow to
filter out the low energy muons. Indeed, only the most energetic muons are
sensitive to temperature variations.

Table 2: Barometric coefficients of the different acquisitions around the blast furnace. Run |  $\beta_{p}$ ($\mathrm{h}\mathrm{P}\mathrm{a}^{-1}$) | Error (RMS) | Correlation coefficient
---|---|---|---
1 | -0.0011 | 1.3$\%$ | 0.77
2 | \- 0.0015 | 0.9$\%$ | 0.80
3 | \- 0.0018 | 1$\%$ | 0.47

A blast furnace is considered to be in operation when air is injected into it,
measured by the value of the so-called blast pressure. At standstill, the
density in the blast furnace is greater and the interior "column" is tighter.
In addition, as the blast pressure increases, fine particles of sinter take
the place of gas and the density in the BF increases. Furthermore, a high coke
fraction in the cohesive zone means that the associated density is lower than
usual. Several parameters can therefore affect the muon flux by changing the
density inside: the fraction of fine particles, the blast furnace stop and the
addition of coke.

We performed multivariate linear fits (with external and internal parameters)
on the relative muon flux and evaluated the adequacy of our fits with the
Pearson linear coefficient of determination. The pressure appears to be the
dominant parameter. Moreover, in October 2021, during the high pressure
period, the coke rate in the cohesive zone seems to be well correlated with
the pressure corrected muon flux. We found $\gamma_{CR}$=-0.013 ($\pm 1\%$)
with a correlation coefficient of 0.76. This means that when the coke rate is
high, the blast furnace is stopped and the material goes down, as well as the
cohesive zone, so the density in the blast furnace increases and the measured
muon flux behind it decreases.

After solving the direct problem we reconstruct the 3D-average-density model
of the blast furnace by using the measured flux (corrected from parameters
that affect it). Markov chain Monte Carlo method discribed in subsection 2.2
is used.

##  4 Results

The results presented in this section were obtained using real data measured
by the detector for three runs.

###  4.1 2D fields

On the figure 2 the opacities (left panels) and densities (right panels) seen
by the detector at each of its 3 positions are represented in 2D, before
inversion. We can see a slightly denser area (in yellow) in the middle of the
figures. This zone would seem to be a 2D projection of the cohesive zone. The
shells of the blast furnace are clearly visible on density representations.
The position 2 density figure shows a denser area in the center left. As
expected, no muons are recorded below 75-90° in the data. Finally, the
contained informations of these figures are aggregated and inverted to
reconstruct the 3D blast furnace and the density distribution inside.

Figure 2: On the left, the opacity (in mwe) and on the right, the density (in
$\mathrm{g}\,\mathrm{c}\mathrm{m}^{-3}$) apparent for each of the 3 positions
are represented. The axes are the azimuth and zenith in ∘.

###  4.2 3D-inversion results

The 3 muon runs were performed at different periods. Therefore, the 3D
reconstruction of the density distribution of the blast furnace gives us an
average of what we can find and not a snapshot of the different zones and
their thickness. In figure 3, the distribution of the mean density (at the
top) and its standard deviation (at the bottom) are shown. Real data have been
inverted by considering the CORSIKA [12] theoretical muon flux model as built
in Cohu et al. [6] for the calculation of flux loss.

  * •

The shell of the blast furnace is clearly visible. It is very dense : more
than 3 $\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$. The travel length from
the detector in position 2 arrive perpendicular to the shell, so we have
directly the value of the density without having to evaluate an integrated
density over the whole width of the blast furnace.

  * •

A brighter area is noticeable on the shell (bottom left of the average density
figures). This is probably an area where the three positions provide different
information on the integrated density. Perhaps one of the detectors cannot
bring measurements from this area, the value of the associated standard
deviations is also high.

  * •

We distinguish, at a height of 15-20 m inside the BF, a sparse zone ($<$0.5
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$) that would contain mainly
coke/coal.

  * •

From this last zone would leave a slightly denser zone in the shape of a
chimney. This phenomenon appears when a lot of coal is pulverized in the
center and little agglomerate. This is the case in the blast furnace that we
have studied.

  * •

The cohesive zone is visible at 20-25 m height, with a density higher than 1.5
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$. It is close to the shell.

  * •

The superposition of materials in the dry zone is not visible. In this zone,
all materials charged from the top of the furnace are in the solid state. The
iron charge and the coke are descending and maintain a layered structure.
There is a superposition of coke and sinter sublayers with a density of 0.7
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$ and 2
$\mathrm{g}\mathrm{/}\mathrm{c}\mathrm{m}^{3}$ respectively. However one can
see some " lines" especially at the top of the image, they are the "travel
length" due to the acceptance effects of the detector.

The results of the 3D inversion obtained here are very satisfactory and the
cohesive zone could be highlighted. The algorithm and the inversion method
have been successfully tested on the internal structure of the BF. We will
test the robustness of the inversion in the next subsection.

###  4.3 Analysis of various sensitivities

We studied, first, the differences observed on our 3D reconstructions as a
function of the muon flux model. We compare Tang et al. [10] and CORSIKA to
calculate the muon flux loss. Tang is an analytical model commonly used and
the CORSIKA flux allows to adapt to the environmental conditions and to the
location of an experiment [6]. We analyzed the performances and uncertainties
of the reconstruction engine : what are the differences caused by the
randomness of the inversion? Then we looked at the consequences of the number
of models registered in the MCMC algorithm and the randomness of the algorithm
itself. We also tested what is implied by a different density model in the
input (different density at the cohesive zone). In all cases, the areas with
the largest density differences are located at the bottom of the blast furnace
: where a few data is collected which is obviously a source of noise.

Figure 3: Results of real data inversion using a theoretical flux modeled with
CORSIKA in Bremen for the calculation of flux loss. The axes ($XYZ$) are in
meters and the density in $\mathrm{g}\,\mathrm{c}\mathrm{m}^{-3}$.
\- at the top : distribution of the average density in the blast furnace,
\- at the bottom : distribution of the standard deviation of the density in
the blast furnace.

##  5 Conclusion

We performed a muography experiment on a blast furnace of ArcelorMittal and
obtained the first 3D image of it. We hoped to be able to clearly distinguish
the location of the cohesive zone, which turns out to be the key of the
furnace’s productivity. The 3D reconstruction of the density distribution was
a great success. We used an inversion program on our measured muon data using
Markov chain Monte Carlo (MCMC). This algorithm is stable (few variations
between 2 identical models) and rather fast even with a large number of
recorded models. The results of the 3D inversion with the use of CORSIKA or
Tang [10] models of theorical flux show that the opacity estimates are
strongly influenced in the regions of zenith angle between 70 and 90° and
especially for the areas of low opacity. As a reminder, the theoretical flux
of CORSIKA is also dependent on atmospheric conditions and must be adapted to
the season when the acquisitions were made. An uncertainty of 10$\%$ on the
flux leads to an error of 4$\%$ on the opacity. The 3D images obtained are
only an average of the density distribution but are quite realistic and
validated by the operators of the blast furnaces studied. The density
contrasts are obvious, especially for the shell and the cohesive zone which
are clearly distinguishable. We could monitor the activity inside the BF.
Indeed, we have evaluated the effect of the atmospheric pressure variation on
the measured muon flux and we are able to correct it for this impact. The
measured and corrected flux seems to be sensitive to the coke rate variations
in the cohesive zone too. All these elements related to the composition or the
shape of the cohesive zone may allow the blast furnace operators to adapt
their material loading according to the state of this zone. Further
improvements to the method are under study.

## Acknowledgments

This work was the subject of a CIFRE agreement between the ArcelorMittal
Maizières Research SA and IP2I (Lyon).

## References

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#  High Throughput Polar Code Decoders with Information Bottleneck
Quantization

Claus Kestel , Lucas Johannsen , and Norbert Wehn  The authors are with the
Microelectronic Systems Design Research Group, RPTU Kaiserslautern-Landau,
67663 Kaiserslautern, Germany (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>The authors acknowledge the
financial support by the Federal Ministry of Education and Research of Germany
in the project “Open6GHub” (grant number: 16KISK004). 0000-0003-3915-9097
0000-0003-1858-8466 0000-0002-9010-086X

###### Abstract

In digital baseband processing, the forward error correction (FEC) unit
belongs to the most demanding components in terms of computational complexity
and power consumption. Hence, efficient implementation of FEC decoders is
crucial for next generation mobile broadband standards and an ongoing research
topic. Quantization has a significant impact on the decoder area, power
consumption and throughput. Thus, lower bit-widths are preferred for efficient
implementations but degrade the error-correction capability. To address this
issue, a non-uniform quantization based on the Information Bottleneck (IB)
method, was proposed that enables a low bit width while maintaining the
essential information. Many investigations on the use of IB method for Low-
density parity-check code (LDPC) decoders exist and have shown its advantages
from an implementation perspective. However, for polar code decoder
implementations, there exists only one publication that is not based on the
state-of-the-art Fast-SSC decoding algorithm, and only synthesis
implementation results without energy estimation are shown. In contrast, our
paper presents several optimized Fast Simplified Successive-Cancellation
(Fast-SSC) polar code decoder implementations using IB-based quantization with
placement&routing results in an advanced 12 nm FinFET technology. Gains of up
to 16 % in area and 13 % in energy efficiency are achieved with IB-based
quantization at a Frame Error Rate (FER) of $10^{-7}$ and a Polar Code of
$N=1024,R=0.5$ compared to state-of-the-art decoders.

###### Index Terms:

Forward Error Correction, Polar Code, Information Bottleneck, ASIC, 12 nm,
Implementation

##  I Introduction

Polar codes are a relatively new class of Forward Error Correction (FEC)
codes, first described by Erdal Arıkan in 2009 [1]. These codes are part of
the 5G standard. They offer low-complexity encoding and decoding algorithms,
which is especially important for high-throughput and low-latency applications
in upcoming standards such as 6G [2]. The most commonly used decoding
algorithms for polar codes, Successive Cancellation (SC) and Successive
Cancellation List (SCL), can be efficiently pipelined to achieve very high
throughput and low latency [3, 4].

Quantization has a significant impact on implementation costs. Coarse
quantization improves implementation efficiency in terms of area, power and
throughput but may decrease the error-correction performance. Finding a good
trade-off is therefore essential for efficient hardware implementations.

One promising technique to maintain the message information but enable a
reduction of the bit width is the Information Bottleneck (IB) method [5].
Here, an information compression is achieved by maximizing the mutual
information between an observed and a compressed random variable for a given
bit width. This yields a non-uniform quantization.

While IB-based quantization for Low Density Parity Check (LDPC) decoder
implementations is well investigated [6, 7, 8], the efficiency of the IB
method for polar code decoder is quite unexplored. It is an open research
question whether IB based quantization in polar decoders can yield more
efficient implementations compared to standard quantization methods. It was
shown in [8, 9], that a pure lookup table (LUT)-based application of the IB
method yields very large LUTs, making this approach unfeasible. The
Reconstruction Computation Quantization (RCQ) scheme [10] also uses LUTs to
reconstruct and, after computation, quantize back to smaller bit width. The
resulting LUTs have much lower complexity. Hence, the RCQ scheme is the most
promising approach from an implementation perspective.

To the best of our knowledge, only one publication investigates the efficiency
of IB-based polar decoder implementation [9]. However, the investigations in
[9] do not consider the state-of-the-art Fast Simplified Successive
Cancellation (Fast-SSC) decoding algorithm and, even more importantly,
provides only synthesis results without any power data, which is one of the
most important implementation metrics.

This work therefore makes the following new contributions:

  * •

We present the first Fast-SSC polar decoder architecture using IB-based
quantization with optimized LUTs to improve area and energy efficiency.

  * •

We compare the non-uniform, IB-based quantization scheme with uniformly
quantized fixed-point (FP) representations in terms of error correction
performance and implementation efficiency for code lengths of
$128\text{\,}\mathrm{bit}\text{/}$ and $1024\text{\,}\mathrm{bit}\text{/}$.

  * •

We analyse the impact of the IB-based quantization on area and power
consumption with 7 different designs in an advanced
$12\text{\,}\mathrm{nm}\text{/}$ FinFET technology.

The remainder of this paper is structured as follows: We provide the required
background of polar codes, their decoding algorithms and the IB method in Sec.
II. IB-based Fast-SSC decoding and our decoder architecture is described in
Sec. III. Sec. IV presents a detailed comparison with uniformly quantized FP
decoders in terms of error correction performance and implementation costs and
Sec. V concludes this paper.

##  II Background

###  II-A Polar Codes

Polar codes $\mathcal{P}(N,K)$ are linear block codes with code length
$N=2^{n}$, that encode $K$ information bits. Channel polarization derives $N$
virtual channels where $K$ reliable channels (information set $\mathcal{I}$)
are used to transmit the information. The $N-K$ remaining (unreliable)
channels are set to zero and called frozen bits (frozen set $\mathcal{F}$).
The encoding includes a bit-reversal permutation [1].

###  II-B Successive Cancellation Decoding

SC decoding can be described as depth-first tree traversal of the Polar Factor
Tree (PFT) [11]. The PFT has ${\log(N)+1}$ stages $s$ and $N$ leaf nodes at
stage $s=0$, representing the frozen and information bits. Each node $v$
receives a Log-Likelihood Ratio (LLR) vector $\boldsymbol{\alpha}^{v}$ of size
$N_{s}$ to first calculate the $N_{s}/2$ elements of the left-child message
$\boldsymbol{\alpha}^{l}$ by the hardware-efficient min-sum formulation of the
$f$-function

$\displaystyle\alpha^{l}_{i}$
$\displaystyle=f(\alpha^{v}_{2i},\alpha^{v}_{2i+1})$ (1)
$\displaystyle=\operatorname{sign}\left(\alpha^{v}_{2i}\right)\operatorname{sign}\left(\alpha^{v}_{2i+1}\right)\min\left(\left|\alpha^{v}_{2i}\right|,\left|\alpha^{v}_{2i+1}\right|\right).$

With the bit vector $\boldsymbol{\beta}^{l}$ received from the left child, the
$N_{s}/2$ elements of $\boldsymbol{\alpha}^{r}$ are calculated using the
$g$-function

$\alpha^{r}_{i}=g\left(\alpha^{v}_{2i},\alpha^{v}_{2i+1},\beta^{l}_{i}\right)=\left(1-2\beta^{l}_{i}\right)\cdot\alpha^{v}_{2i}+\alpha^{v}_{2i+1},$
(2)

and sent to the child on the right. With the results of both children, the
$h$-function calculates the partial-sum $\boldsymbol{\beta}^{v}$ with
$\oplus$, the binary XOR-operator, as

$\left(\beta^{v}_{2i},\beta^{v}_{2i+1}\right)=h\left(\beta^{l}_{i},\beta^{r}_{i}\right)=\left(\beta^{l}_{i}\oplus\beta^{r}_{i},\beta^{r}_{i}\right),$
(3)

that is sent to the parent node. In leaf nodes, bit decisions are made. Frozen
bits are $0$ per definition, and information bit nodes return the Hard
Decision (HD) on the LLR:

$\beta^{v}=\operatorname{sign}\left(\alpha^{v}\right)\triangleq\begin{cases}0&\text{if
}\alpha^{v}\geq 0\\\ 1&\text{otherwise.}\end{cases}$ (4)

The decoder outputs the partial-sum $\boldsymbol{\beta}^{0}$ of the root node.

###  II-C Fast-SSC Decoding

Pruning the PFT reduces the number of operations required to decode one code
word [11]. Subtrees containing only frozen bits do not have to be traversed,
because their decoding result is known to be an all-zero vector in advance.
Such Rate-0 nodes are always left children and are merged into their parent
nodes. Here, the $g$\- and $h$-functions are executed with the known all-zero
$\boldsymbol{\beta}^{l}$, denoted by $g_{0}$ and $h_{0}$, respectively.
Similarly, subtrees without any frozen bits can be decoded directly by the HD,
because no parity information is contained. Merging these Rate-1 nodes results
in the $h_{1}$-function, which directly calculates $\boldsymbol{\beta}^{r}$
using

$\beta^{r}_{i}=\operatorname{sign}\left(g\left(\alpha^{v}_{2i},\alpha^{v}_{2i+1},\beta^{l}_{i}\right)\right),$
(5)

and combines it with $\boldsymbol{\beta}^{l}$ to observe
$\boldsymbol{\beta}^{v}$.

Fast-SSC decoding [12] applies further optimizations: If a subtree contains
only one information bit, it is considered a Repetition (REP) code and
replaced by a specialized REP node. All its bits are decoded by summing up the
vector $\alpha^{v}$ of received LLR values and extracting the sign bit of the
sum:

$\beta^{v,\text{REP}}_{i}=\operatorname{sign}\left(\sum_{j=0}^{N_{s}-1}\alpha^{v}_{j}\right).$
(6)

In subtrees containing only one frozen bit, this bit always acts as parity
bit. Thus, the partial sum of this subtree represents a Single Parity-Check
(SPC) code. A specialized SPC node performs Maximum Likelihood (ML) decoding
by calculating the parity $\gamma^{v}\in\\{0,1\\}$ of the input:

$\gamma^{v}=\bigoplus_{i=0}^{N_{s}-1}\operatorname{sign}\left(\alpha^{v}_{i}\right),$
(7)

finding the least reliable bit

$i_{\min}=\operatorname*{arg\,min}_{i\in[0,N_{s})}\left|\alpha^{v}_{i}\right|$
(8)

and setting $\boldsymbol{\beta}^{v}$ to satisfy the single parity constraint:

$\beta^{v,\text{SPC}}_{i}=\begin{cases}\operatorname{sign}\alpha^{v}_{i}\oplus\gamma^{v}&\text{if
}i=i_{\min}\\\ \operatorname{sign}\alpha^{v}_{i}&\text{otherwise.}\end{cases}$
(9)

###  II-D Information Bottleneck Method

The IB method is a mathematical framework used for clustering in information
theory and machine learning [5]. In the IB setup, the target is to preserve
the shared mutual information $I(X;Y)$ between an observed random variable $Y$
and the relevant random variable $X$ while compressing $Y$ to $T$, i. e.,
maximizing $I(T;X)$. Different IB algorithms exist [13] to provide the
compression mapping $p(t|y)$ derived only from the joint distribution $p(x;y)$
and the cardinality of the compressed event space ($|\mathcal{T}|$), with
$x\in\mathcal{X}=\\{0,1\\},y\in\mathcal{Y},t\in\mathcal{T}$ being realizations
of the random variables $X,Y,T$, respectively and
$|\mathcal{T}|\ll|\mathcal{Y}|$. A collection of IB algorithms is provided by
[14] and is used in this work. This compression is applied to the output of an
Additive White Gaussian Noise (AWGN) channel to obtain a coarse, non-uniform
quantization.

In the case of our IB decoder, $y\in\mathcal{Y}$ are the received LLRs that
are quantized with a high bit width, e.g.
$10\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$, which equals
$|\mathcal{Y}|=2^{10}=1024$ bins. The IB algorithm then iteratively tries to
find pairs of bins to combine into one bin with the least loss of mutual
information $I(X;Y)$. In that way $|\mathcal{Y}|$ is reduced to
$|\mathcal{T}|=16$ ($4\text{\,}\mathrm{b}\mathrm{i}\mathrm{t}$) and a mapping
from $y$ to $t$ is derived.

##  III IB Decoder

###  III-A Numerical Representations and LUT Generation

In this paper, we focus on decoders with very high throughput and low latency.
These decoders are fully unrolled and pipelined and use a Sign-Magnitude (SM)
representation of the quantized, FP LLR values [15]. For high Signal-to-Noise-
Ratios (SNRs), saturation reduces signal toggling because only the sign bit
changes, which reduces power consumption. Additionally, the comparators in the
$f$-functions (1) and SPC nodes (8) can directly operate on the magnitude. To
perform the additions in the $g$-functions (2) and REP nodes (6), the SM
representations are converted to Two’s-Complement (TC) to efficiently perform
the calculations.

Our IB decoder implementations exploit all these optimizations. However,
because of the non-uniform distribution characterizing the IB indices,
mathematical computations must be replaced by LUTs [9]. However, these LUTs
can become extremely large. For example, for a $g$-function (2) with one
binary and two LLR inputs, the resulting LUT is of size
$2\cdot|\mathcal{T}|^{2}$. From an implementation perspective, the LUTs are
transformed into Boolean functions. Despite the logic optimization executed by
state-of-the-art synthesis tools, the resulting logic costs can quickly
outweigh the benefits of reduced bit widths, particularly for increasing LUT
sizes [8].

A promising approach to address this problem is the RCQ scheme [10, 16]. RCQ
combines non-uniform quantization with the traditional node computations. Only
small LUTs are necessary that are placed in front of the computation units to
upscale the reduced non-uniform IB quantization to a uniform FP quantization
(_Reconstruction_). Node computations are then performed on the uniform FP
quantization (_Computation_). After computation the results are downscaled to
smaller bit width (_Quantization_). The RCQ scheme results in much smaller
LUTs because the mappings for the conversions between IB and FP domains can be
implemented as LUTs of size $2^{Q}$ for each value where $Q$ is the bit width.
We use $Q_{\text{IB}}=\log_{2}(|\mathcal{T}|)$ and $Q_{\text{FP}}$ to denote
the bit widths in the IB and FP domains, respectively. For the $g$-function
example mentioned above, the number of entries in the LUTs shrinks from
$2\cdot|\mathcal{T}|^{2}=2^{1+2\cdot Q_{\text{IB}}}$ to $2\cdot
2^{Q_{\text{IB}}}+2^{Q_{\text{FP}}}$.

TABLE I:  Hardware-aware representation of $3\text{\,}\mathrm{bit}\text{/}$ IB indices  IB Index $t$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7
---|---|---|---|---|---|---|---|---
SM IB | -3 | -2 | -1 | -0 | 0 | 1 | 2 | 3
Binary | 1 11 | 1 10 | 1 01 | 1 00 | 0 00 | 0 01 | 0 10 | 0 11

We use density evolution to generate samples. At least 100K samples are
monitored at each node in the decoder, i. e., at each edge of the PFT. These
samples yield the joint distributions $p(x;y)$ which are input to the IB
algorithm [13] that calculates the compression mappings $p(t|y)$ for every
edge. Then, the Symmetric information bottleneck algorithm [7, 14] is applied
which is optimized to preserve the symmetry of the transmission channel
(assuming an AWGN channel and Binary Phase Shift Keying (BPSK) modulation).
Exploiting this symmetry enables a bisection of the LUT , because it is
sufficient to store only the magnitudes. Accordingly, we use a SM-like
representation of the IB indices as shown in the example in Table I for
$|\mathcal{T}|=8$. Therefore, for both directions of conversion (IB to FP, and
FP to IB), the LUTs directly map one magnitude to another, i. e., the
magnitudes in both domains also act as indices for the LUTs. Thus, the size of
each LUT is $2^{Q-1}$ and the total number of entries for the example of the
$g$-function becomes $2\cdot 2^{Q_{\text{IB}}-1}+2^{Q_{\text{FP}}-1}$. As
shown in Figure 1 this means a reduction from $2*16^{2}=512$ entries to
$2*2^{4-1}$ for the upscaling LUT and $2^{5-1}$ for the downscaling LUT,
making it just 32 entries.

Furthermore, this approach eliminates the need for multiple comparisons with
thresholds per conversion as in [16].

#### Notation

To differentiate between the numerical representations, we use $\alpha$ to
denote values in the IB domain, $\tilde{\alpha}$ for SM and $\ddot{\alpha}$
for TC representation. The $j$-th bit of the binary expansion of $\alpha_{i}$
is given by $\alpha_{i_{(j)}}$ and the most significant bit (MSB) $Q-1$ refers
to the sign bit.

###  III-B IB-based Fast-SSC Decoding

####  III-B1 $f$-Function

With the symmetric mapping and inherent order of the IB indices (Table I), the
$f$-function (1) can be directly performed in the non-uniform IB domain and no
LUTs are necessary, which corresponds to the “ _re-MS-IB_ decoder”
implementation of [9]. In contrast to [9], we map the IB indices so that
negative values also correspond to negative LLRs and, thus, do not need to
invert the result of the _XOR_ -ed sign bits.

####  III-B2 $g$-Function

$\alpha^{r}_{0_{(3)}}$ $\alpha^{r}_{0_{(2)}}$ $\alpha^{r}_{0_{(1)}}$
$\alpha^{r}_{0_{(0)}}$ $\tilde{\alpha}^{r}_{0_{(4)}}$
$\tilde{\alpha}^{r}_{0_{(3)}}$ $\tilde{\alpha}^{r}_{0_{(2)}}$
$\tilde{\alpha}^{r}_{0_{(1)}}$ $\tilde{\alpha}^{r}_{0_{(0)}}$
$\ddot{\alpha}^{r}_{0_{(5)}}$ $\ddot{\alpha}^{r}_{0_{(4)}}$
$\ddot{\alpha}^{r}_{0_{(3)}}$ $\ddot{\alpha}^{r}_{0_{(2)}}$
$\ddot{\alpha}^{r}_{0_{(1)}}$ $\ddot{\alpha}^{r}_{0_{(0)}}$
$\sum$$\ddot{\alpha}^{v}_{0_{(4)}}$ $\ddot{\alpha}^{v}_{0_{(3)}}$
$\ddot{\alpha}^{v}_{0_{(2)}}$ $\ddot{\alpha}^{v}_{0_{(1)}}$
$\ddot{\alpha}^{v}_{0_{(0)}}$ $\ddot{\alpha}^{v}_{1_{(4)}}$
$\ddot{\alpha}^{v}_{1_{(3)}}$ $\ddot{\alpha}^{v}_{1_{(2)}}$
$\ddot{\alpha}^{v}_{1_{(1)}}$ $\ddot{\alpha}^{v}_{1_{(0)}}$
$\tilde{\alpha}^{v}_{0_{(4)}}$ $\tilde{\alpha}^{v}_{0_{(3)}}$
$\tilde{\alpha}^{v}_{0_{(2)}}$ $\tilde{\alpha}^{v}_{0_{(1)}}$
$\tilde{\alpha}^{v}_{0_{(0)}}$ $\tilde{\alpha}^{v}_{1_{(4)}}$
$\tilde{\alpha}^{v}_{1_{(3)}}$ $\tilde{\alpha}^{v}_{1_{(2)}}$
$\tilde{\alpha}^{v}_{1_{(1)}}$ $\tilde{\alpha}^{v}_{1_{(0)}}$
$\alpha^{v}_{0_{(3)}}$ $\alpha^{v}_{0_{(2)}}$ $\alpha^{v}_{0_{(1)}}$
$\alpha^{v}_{0_{(0)}}$ $\alpha^{v}_{1_{(3)}}$ $\alpha^{v}_{1_{(2)}}$
$\alpha^{v}_{1_{(1)}}$ $\alpha^{v}_{1_{(0)}}$ $\beta^{l}_{0}$ $Q_{\text{FP}}=$
$5\text{\,}\mathrm{bit}\text{/}$ (SM)$Q_{\text{IB}}=$
$4\text{\,}\mathrm{bit}\text{/}$ (IB)$Q_{\text{FP}}=$
$5\text{\,}\mathrm{bit}\text{/}$ (TC)$Q_{\text{FP}}=$
$6\text{\,}\mathrm{bit}\text{/}$ (TC)$Q_{\text{FP}}=$
$5\text{\,}\mathrm{bit}\text{/}$ (SM)$Q_{\text{IB}}=$
$4\text{\,}\mathrm{bit}\text{/}$
(IB)$\mathrm{LUT}_{\text{up}}$$\mathrm{LUT}_{\text{up}}$conversionconversionsaturation
&conversion$\mathrm{LUT}_{\text{down}}$

Figure 1: RCQ schematic for the $g$-function

As in [16], we apply an RCQ scheme, but based on our optimized up- and
downscale LUTs:

$\alpha^{r}_{i}=\mathrm{LUT}^{v}_{\text{down}}\left(g\left(\mathrm{LUT}^{v}_{\text{up}}\left(\alpha^{v}_{2i}\right),\mathrm{LUT}^{v}_{\text{up}}\left(\alpha^{v}_{2i+1}\right),\beta^{l}_{i}\right)\right).$
(10)

The internal separation between the different number representations is
maintained for the reasons described in Sec. III-A and shown in Fig. 1. The
_Reconstruction_ with $\mathrm{LUT}^{v}_{\text{up}}$ maps the magnitude of the
IB index $\alpha^{v}_{i}$ to its SM representation $\tilde{\alpha}^{v}_{i}$
with preserved sign bit. $\beta^{l}_{i}$ must be considered before conversion
to the TC representation $\ddot{\alpha}^{v}_{i}$ for the _Computation_ step.
The result $\ddot{\alpha}^{r}_{i}$ has a $1\text{\,}\mathrm{bit}\text{/}$
larger resolution, implying a saturation for the conversion back to SM
representation with $Q_{\text{FP}}\text{\,}\mathrm{bit}\text{/}$. The
_Quantization_ step is again realized as magnitude
$\mathrm{LUT}^{v}_{\text{down}}$ for the transformation back to the IB domain.

For the special case of merged Rate-0 nodes, i. e., the $g_{0}$-function, the
_XOR_ -operation with $\beta^{v}_{i}$ is omitted.

####  III-B3 REP Nodes

REP nodes calculate the sum over all input values to observe the single
(repeated) information bit by HD on the sum (6). Thus, the RCQ scheme as
described for the $g$-function is applied for REP nodes. An adder tree of
$N_{s}-1$ adders and a depth of $s=\log_{2}N_{s}$ operates on the TC
representations of the $N_{s}$ input values. The final HD as _Quantization_
step extracts the single sign bit as bit decision of the node, for which
reason no further conversion of the sum is needed.

####  III-B4 SPC Nodes

As in the f-function, due to the ordered mapping of the IB indices, the
minimum search of the SPC node (8) can be performed directly in the IB domain.
Furthermore, the chosen mapping is also suitable for the direct parity
calculation (7) and the bit estimations (9), because the sign bits are
preserved in the IB domain.

####  III-B5 $h_{1}$-Function

The $h_{1}$-function internally uses the $g$-function to compute $\beta^{r}$
(5). However, in this $g$-function, the backward conversion is not needed,
because the HD is made directly in the compute domain with TC representation,
as already described for the REP nodes.

###  III-C Decoder Architecture

FG0SPCH0GFREPH1H$\boldsymbol{\alpha}^{0}$$\boldsymbol{\alpha}^{1}$$\boldsymbol{\alpha}^{4}$$\boldsymbol{\alpha}^{2}$$\boldsymbol{\alpha}^{5}$$\boldsymbol{\beta}^{4}$$\boldsymbol{\beta}^{1}$$\boldsymbol{\beta}^{5}$$\boldsymbol{\beta}^{2}$$\boldsymbol{\beta}^{0}$$\boldsymbol{\alpha}^{0}$$\boldsymbol{\alpha}^{0}$$\boldsymbol{\alpha}^{0}$$\boldsymbol{\alpha}^{0}$$\boldsymbol{\alpha}^{2}$$\boldsymbol{\alpha}^{2}$$\boldsymbol{\beta}^{1}$$\boldsymbol{\beta}^{1}$$\boldsymbol{\beta}^{1}$$\boldsymbol{\beta}^{1}$

Figure 2:  Unrolled and pipelined Fast-SSC decoder architecture for a
$\mathcal{P}(16,8)$. Colors represent numerical domains: green for IB, red for
FP, blue for binary and dark green shows the LUTs.

An outline of the fully unrolled and deeply pipelined Fast-Simplified SC (SSC)
decoder architecture for a $\mathcal{P}(16,8)$ is shown in Figure 2. We omit
the clock signals and the delay lines are represented by shift registers. The
pipeline consists of various building blocks that implements the decoding
functions described in the previous section. The IB, FP and binary domains are
represented by the coloring of the blocks and signals. The decoders presented
in this paper are based on the framework presented in [15] which we extended
to apply the IB method as described above.

##  IV Results

We present 7 decoder designs for $N\\!=\\!128$ and $N\\!=\\!1024$ which are
optimized for a target frequency of $500\text{\,}\mathrm{MHz}\text{/}$ and
$750\text{\,}\mathrm{MHz}\text{/}$, respectively. Throughput is considered as
coded throughput. The designs were synthesized with Design Compiler and placed
and routed with IC-Compiler, both from Synopsys in a
$12\text{\,}\mathrm{nm}\text{/}$ FinFET technology with a super-low $V_{t}$
transistor library from Global Foundries under worst case Process, Voltage and
Temperature (PVT) conditions
($125\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$,
$0.72\text{\,}\mathrm{V}\text{/}$) for timing and nominal case PVT
($25\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$,
$0.8\text{\,}\mathrm{V}\text{/}$) for power. Error-correction performance was
simulated for an AWGN channel and BPSK modulation with a minimum of 100
erroneous code words.

###  IV-A Decoder for the Polar(128,64) Code

The Frame Error Rate (FER) of the $\mathcal{P}(128,64)$ is shown in Figure 3.
The IB decoder with $4\text{\,}\mathrm{bit}\text{/}$ and the FP decoder with
$5\text{\,}\mathrm{bit}\text{/}$ show similar error-correction performance,
whereas the error-correction performance of the
$4\text{\,}\mathrm{bit}\text{/}$ FP decoder starts to degrade at an FER of
$10^{-3}$.

TABLE II:  Implementation results for $\mathcal{P}(128,64)$ decoders

|  IB 4bit |  [9]* |  FP 4bit |  FP 5bit
---|---|---|---|---
|  Place&Route |  Synthesis |  Place&Route |  Place&Route
Technology |  $12\text{\,}\mathrm{nm}\text{/}$ |  $28\text{\,}\mathrm{nm}\text{/}$$\rightarrow$$12\text{\,}\mathrm{nm}\text{/}$ |  $12\text{\,}\mathrm{nm}\text{/}$ |  $12\text{\,}\mathrm{nm}\text{/}$
Frequency [MHz] |  500 |  1000 (3681) |  500 |  500
Throughput [Gbps] |  64 |  13 (47) |  64 |  64
Latency [ns] |  18 |  86 (23) |  18 |  18
Latency [CC] |  9 |  86 |  9 |  9
Area [mm2] |  0.014 |  0.026 |  0.012 |  0.015
\- Registers |  0.005 |  — |  0.005 |  0.006
\- Combinatorial |  0.005 |  — |  0.003 |  0.005
Area Eff. [Gbps/mm2] |  4528 |  502 (1847) |  5384 |  4248
Power Total [mW] |  21 |  — |  17 |  24
\- Clock |  7 |  — |  7 |  9
\- Registers |  3 |  — |  3 |  4
\- Combinatorial |  10 |  — |  7 |  11
Energy Eff. [pJ/bit] |  0.32 |  — |  0.27 |  0.37
Power Density [W/mm2] |  1.46 |  — |  1.47 |  1.58

* Synthesis only, scaled from $28\text{\,}\mathrm{nm}\text{/}$ to $12\text{\,}\mathrm{nm}\text{/}$ (numbers in brackets) with equations from [17] and maximum frequency limited to $1000\text{\,}\mathrm{MHz}\text{/}$. For $28\text{\,}\mathrm{nm}\text{/}$ and $12\text{\,}\mathrm{nm}\text{/}$ the scaling factors of $32\text{\,}\mathrm{nm}\text{/}$ and $14\text{\,}\mathrm{nm}\text{/}$ were used, since they belong to the same technology generation and give the best approximation.

Figure 3: FER of the $\mathcal{P}(128,64)$, Float vs. FP vs. IB Quantization

Table II presents the corresponding implementation results. Comparing the
decoders with similar FER (IB vs. $5\text{\,}\mathrm{bit}\text{/}$ FP), we
observe similar combinatorial area (logic), whereas the area for the memory
(registers) is reduced by ~$16\text{\,}\mathrm{\char 37\relax}\text{/}$. This
improves the area efficiency by ~$7\text{\,}\mathrm{\char 37\relax}\text{/}$
and yields power savings and energy efficiency improvements of
~$13\text{\,}\mathrm{\char 37\relax}\text{/}$.

When comparing the IB decoder with the $4\text{\,}\mathrm{bit}\text{/}$ FP
decoder, the cost of the LUTs can be directly observed in the combinatorial
area (0.005 mm2 vs. 0.003mm2) while the area for the registers is identical.
This cost can be considered as the price for the improved error correction
performance of the IB decoder.

As already mentioned there exits only one other publication that gives
implementation results for IB-based decoders. Since the authors of [9] only
provide synthesis results in an older $28\text{\,}\mathrm{nm}\text{/}$
technology, a fair comparison is difficult. To enable at least some
comparison, we scaled the results of [9] to $12\text{\,}\mathrm{nm}\text{/}$
according to the equations of [17]. The scaled results are included in Table
II. We limited the maximum frequency to $1000\text{\,}\mathrm{MHz}\text{/}$
which is more reasonable, since $3681\text{\,}\mathrm{MHz}\text{/}$ (and even
$1510\text{\,}\mathrm{MHz}\text{/}$ in the original publication) are not
realistic for a design in $28\text{\,}\mathrm{nm}\text{/}$ after placement.
The reasons are: first, the power consumption and power density becomes
infeasible with this high frequency. Second,
$3681\text{\,}\mathrm{MHz}\text{/}$ are unfeasible for standard placement and
routing in semi-custom design flows in $28\text{\,}\mathrm{nm}\text{/}$
technology. Even with the scaled estimate without frequency limitation, we
observe that our optimised decoders outperform [9] in throughput, latency,
area and area efficiency.

###  IV-B Decoder for the Polar(1024,512) Code

The longer $\mathcal{P}(1024,512)$ polar code has a longer pipeline and is
therefore more affected by accumulating quantization errors. In contrast to
the shorter code, $6\text{\,}\mathrm{bit}\text{/}$ FP are necessary to match
the floating point precision. We compare the $4\text{\,}\mathrm{bit}\text{/}$
IB decoder to the FP decoder with $5\text{\,}\mathrm{bit}\text{/}$, as they
show similar error-correction performance (Figure 4). Here, the IB decoder
even outperforms the $5\text{\,}\mathrm{bit}\text{/}$ FP decoder at an FER of
$10^{-7}$.

Figure 4:  FER of the $\mathcal{P}(1024,512)$, Float vs. FP vs. IB TABLE III:  Implementation results for $\mathcal{P}(1024,512)$ decoders | IB 4bit | FP 4bit | FP 5bit | FP 6bit
---|---|---|---|---
Frequency [MHz] | 750 | 750 | 750 | 750
Throughput [Gbps] | 768 | 768 | 768 | 768
Latency [CC] | 92 | 92 | 92 | 92
Area [mm2] | 0.822 | 0.785 | 0.968 | 1.158
\- Registers | 0.360 | 0.360 | 0.439 | 0.517
\- Combinatorial | 0.200 | 0.168 | 0.208 | 0.274
Area Eff. [Gbps/mm2] | 935 | 979 | 794 | 663
Power Total [W] | 0.984 | 0.866 | 1.149 | 1.431
\- Clock | 0.208 | 0.207 | 0.254 | 0.298
\- Registers | 0.280 | 0.263 | 0.354 | 0.404
\- Combinatorial | 0.479 | 0.381 | 0.523 | 0.706
Energy Eff. [pJ/bit] | 1.28 | 1.13 | 1.50 | 1.86
Power Density [W/mm2] | 1.20 | 1.10 | 1.19 | 1.24

Comparing the implementation results of the IB decoder and the
$5\text{\,}\mathrm{bit}\text{/}$ FP decoder (Table III), the total area is
reduced by ~$15\text{\,}\mathrm{\char 37\relax}\text{/}$, mostly stemming from
the reduction in registers. This leads to an improved area efficiency of
~$18\text{\,}\mathrm{\char 37\relax}\text{/}$, whereas the energy efficiency
improves by ~$15\text{\,}\mathrm{\char 37\relax}\text{/}$. Figure 5 shows the
layouts of the $5\text{\,}\mathrm{bit}\text{/}$ FP and the
$4\text{\,}\mathrm{bit}\text{/}$ IB decoder. It is also worth noting that,
when compared to the close-to-float $6\text{\,}\mathrm{bit}\text{/}$ FP
decoder, accepting a small loss of ~$0.2\text{\,}\mathrm{dB}\text{/}$ in the
error correction leads to improvements of ~$41\text{\,}\mathrm{\char
37\relax}\text{/}$ in area efficiency and ~$31\text{\,}\mathrm{\char
37\relax}\text{/}$ energy efficiency.

(a) FP $5\text{\,}\mathrm{bit}\text{/}$:
$0.968\text{\,}{\mathrm{mm}}^{2}\text{/}$

(b) IB $4\text{\,}\mathrm{bit}\text{/}$:
$0.822\text{\,}{\mathrm{mm}}^{2}\text{/}$

Figure 5: Layout pictures for FP and IB polar code decoders

##  V Conclusion

We presented fully characterized Fast-SSC Polar Decoders with an optimized IB-
based quantization scheme. Especially for ultra-high throughput we outperform
decoders with comparable bit-width by $18\text{\,}\mathrm{\char
37\relax}\text{/}$ in area efficiency and $15\text{\,}\mathrm{\char
37\relax}\text{/}$ in energy efficiency. This effect can be explained by the
savings in memory requirements of fully pipelined and unrolled decoders which
is minimized with the IB-based quantization.

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