Partial Orders in Rate-Matched Polar Codes

Zhichao Liu^∗†, Liuquan Yao^∗†, Yuan Li^‡, Huazi Zhang^‡, Jun Wang^‡, Guiying Yan^∗† and Zhiming Ma^∗†
^∗University of Chinese Academy and Sciences, Beijing, China
^†Academy of Mathematics and Systems Science, CAS, Beijing, China
^‡Huawei Technologies Co. Ltd., China
Email: {liuzhichao20, yaoliuquan20}@mails.ucas.ac.cn, {liyuan299, zhanghuazi, justin.wangjun}@huawei.com,
yangy@amss.ac.cn, mazm@amt.ac.cn This work was supported by the National Key R&D Program of China (No. 2023YFA1009602).

Abstract

In this paper, we establish the partial order (POs) for both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC) under any block rate-matched polar codes. Firstly, we define the POs in the sense of rate-matched polar codes as a sequential block version. Furthermore, we demonstrate the persistence of POs after block rate matching in the BEC. Finally, leveraging the existing POs in the BEC, we obtain more POs in the BMSC under block rate matching. Simulations show that the PW sequence constructed from $\beta$ -expansion can be improved by the tool of POs. Actually, any fixed reliable sequence in the mother polar codes can be improved by POs for rate matching.

Index Terms:

Polar Codes, Sequential Rate-matched, Partial Order.

I Introduction

Since Arıkan’s introduction of polar codes [1], polar codes have garnered significant attention and research interest, which are not only capacity achieving, but also have encoding and decoding algorithm with low complexity. In traditional polar codes, as the code length $N$ approaches infinity, the ratio of perfect channels converges to the channel capacity $I(W)$ , while the ratio of purely noisy channels approaches $1-I(W)$ . Thus, a crucial aspect in polar code research is the selection of the $K$ most reliable bit channels from $N$ synthetic channels to carry information bits. For traditional polar codes, there are many methods to select information bits, such as Gaussian approximation (GA) algorithm [2], PW construction algorithm [3] and 5G sequence [4].

In engineering applications, the code length $N$ typically does not reach exponential powers. To address this, the issue of rate matching has been introduced in [5] and [6]. Generally, a bit-puncturing strategy is employed at lower rates, while a bit-shortening strategy is employed at higher rates. Recently, it has been demonstrated that puncturing and shortening polar codes achieve capacity in [7]. There have been many studies related to rate matching and refer to [7]-[14] for more details. Although we can use algorithms such as Gaussian approximation to reconstruct the bit reliability of the rate-matched polar codes, the corresponding complexity is still very high. While if we can obtain a family of POs, we can avoid some repeated reliable calculations under different SNR. Therefore, it remains highly significant to investigate the POs of rate-matched polar codes.

Because the analysis of the Bhattacharyya parameters has recursive expressions for BEC, many researches study the POs in the BEC [15] [16] [17]. Channel degradation has been proposed in [18], then in [19] they verify the partial order between $10$ and $01$ under channel degradation by a sufficient condition. What’s more, [20] studies the generalized partial order of BEC and BMSC. However, the analysis after rate matching in the BEC and BMSC is complex because the reliability of positions changes without recursion. Despite its engineering significance, this issue has received little attention. As we know, we firstly propose the POs for both the BEC and the BMSC under sequential rate-matched polar codes.

In [21], we explored new POs for traditional polar codes by analyzing the upper and lower bounds of the Bhattacharyya parameters in BMSC. We find more POs defines by Bhattacharyya parameter and error probability, which are richer than POs in the sense of channel degradation. Building upon this work, our objective is to find more POs in the BEC and BMSC under sequentially punctured polar codes.

I-A Contributions

The contributions of our paper are summarized as follows:

1.

We firstly propose a study of partial order under block rate matching, and define the partial order of BEC and BMSC under block rate matching. This has a certain enlightenment effect on the ranking of the reliability of polar codes after rate matching.
2.

We extend some inherited POs, and propose a sufficient condition for verifying the inherited POs of the BEC under block rate matching. Since the partial order of the mother polar code under the BEC is rich, this research provides a certain guarantee for the diversity of the partial order of the BEC under rate matching.
3.

We introduce novel POs for the BMSC under block rate matching. Our conclusion leverages the existing POs for BEC under block rate matching to derive new POs applicable to arbitrary BMSC also under the same block rate matching. The proof constitutes a central challenge of our work. In order to obtain an inequality of the Bhattacharyya parameters of the synthesized channel after rate matching, our approach constructs a convolution mapping and demonstrates that the geometric mean pair exhibits a superior polarization effect compared to the uneven pair of initial Bhattacharyya parameters.

We have listed the main conclusions of our paper in Table I.

TABLE I: Inherited BEC POs and POs for BMSC

Conditions	POs
$a\preceq_{P,m,BEC}b,\|a\|,\|b\|\geq m$	$ac\preceq_{P,m,BEC}bc$
$a\preceq_{P,m,BEC}b,c\preceq_{BEC}d,\|a\|,\|b\|\geq m$	$ac\preceq_{P,m,BEC}bd$
$n,t,\ell\geq 0$	$p^{n}0r^{\ell}1q^{t}\preceq_{P,m,BEC}p^{n}1r^{\ell}0q^{t}$
$0^{p}1^{m-p}\alpha\preceq_{BEC}0^{q}1^{m-q}\gamma$ , $q\leq p$	$0^{p}1^{m-p}\alpha\preceq_{1,m,BEC}0^{q}1^{m-q}\gamma$
$\mid\gamma\mid=m-1$ , $\gamma 0\alpha\preceq_{BEC}\gamma 1\beta$	$\gamma 0\alpha\preceq_{P,m,BEC}\gamma 1\beta$
$\gamma\preceq_{P,m,BEC}\alpha$	$\gamma 1\preceq_{P,m,BMSC}1\alpha$

I-B Organizations and Notations

This paper is organized as follows. We review the definition of polar codes and rate matching, and define the partial order under block rate matching in Section II. In Section III, we give some inherited POs in the BEC and a sufficient condition for block rate matching. In Section IV, we give the details of the proof of the new POs in BMSC along with an illustrative example. Simulation results are presented in Section V. Finally, conclusions are drawn in Section VI.

Greek letters denote the paths of polarization, where ’ $0$ ’ represents up polarization and ’ $1$ ’ represents down polarization. The modulo length $|\cdot|$ indicates the length of the path.

For example, $\alpha=1100$ represents a path involving two down polarization transformations followed by two up polarization transformations. We use $W^{\alpha}$ to denote the synthesized channel generated by the polarization path $\alpha$ with the initial channel $W$ . In this paper, we exclusively consider $W$ as either a BEC or BMSC.

For consistent initial input variables, we define the up and down polarization functions as follows: $f_{0}(x):=1-(1-x)^{2},f_{1}(x):=x^{2}$ with $x\in[0,1]$ , and then for a path $\alpha$ with length $n$ , we define

	$\displaystyle f_{\alpha}(x)$	$\displaystyle=f_{\alpha_{1}\alpha_{2}\cdots\alpha_{n}}(x)$		(1)
		$\displaystyle:=f_{\alpha_{n}}\circ f_{\alpha_{n-1}}\circ\cdots\circ f_{\alpha_{1}}(x),\;\;x\in[0,1].$		(1)

Generally, we define up and down polarization operations as ’ $\bar{*}$ ’ and ’ $\underline{*}$ ’ relatively for different input variables:

\left\{\begin{aligned} &a\bar{*}b=a+b-ab,\\ &a\underline{*}b=ab.\\ \end{aligned}\right.

(2)

For a vector $Z$ of length $N=2^{n}$ , where $Z=(z_{1},\cdots,z_{n})$ , we denote the polarized vector of $Z$ as $h(Z)$ . For example, when $Z=(z_{1},z_{2},z_{3},z_{4})$ , then

	$\displaystyle h(Z)=$	$\displaystyle[(z_{1}\bar{}z_{3})\bar{}(z_{2}\bar{}z_{4}),(z_{1}\bar{}z_{3})\underline{}(z_{2}\bar{}z_{4}),$		(3)
		$\displaystyle(z_{1}\underline{}z_{3})\bar{}(z_{2}\underline{}z_{4}),(z_{1}\underline{}z_{3})\underline{}(z_{2}\underline{}z_{4})]$		(3)

Fig. 1 illustrates the polarized process of different input variables.

Refer to caption — Figure 1: $h(Z)$ as the polarized vector of $Z$ for $N=4$ .

$h(Z_{1})\leq h(Z_{2})$ indicates that every component of the vector satisfies the inequality.

II Preliminaries

II-A Polar Codes

Let $N=2^{n}$ , $F_{N}=B_{N}F_{2}^{\otimes n}$ , where $F_{2}=\left(\begin{aligned} &1&0\\ &1&1\\ \end{aligned}\right)$ and $B_{N}$ is the bit-reversal permutation matrix. Then a $\mathcal{P}(N,K)$ polar code can be generated by choosing $K$ rows in $F_{N}$ as the information bit set $\mathcal{I}$ , and other rows are denoted by $\mathcal{F}$ . The transmitted code can be encoded by $\bm{x}=\bm{u}F_{N}$ , where only $\bm{u}(\mathcal{I})$ can carry information while $\bm{u}(\mathcal{F})=0$ . The codeword $\bm{x}$ is transmitted through a channel and successive cancellation (SC) decoder [1] is a frequently used decoder algorithm with low complexity $O(NlogN)$ . In order to improve the performance of decoder, a successive cancellation list (SCL) decoder is proposed in [22]. And a SCL decoder with CRC is studied in [23] for detecting the error in the list.

II-B Rate Matching

The code length of the polar code is an integer power of two: $N=2^{n}$ , but the code length is usually required to be an arbitrary positive integer. Therefore, rate matching is a method used to modify the code length and adjust rate. The topics we consider for rate matching include block puncturing and sequential shortening.

1.

$Puncturing:$ Puncturing makes $\bar{P}$ bits incapable to modify the code length, and the punctured code length is $N-\bar{P}$ . After the decoder receives the received vector, it has no punctured bit information at all. Thus, the LLRs of the punctured bit is $0$ and the Bhattacharyya parameter is $1$ . Block puncturing means sequentially puncturing out the encoded $1,2,\cdots,\bar{P}$ bits.
2.

$Shortening:$ Shortening makes $\bar{S}$ bits fixed, and when the decoder receives the received vector, it is fully aware of the information about the shortened bits. Therefore, the LLR of the shortened bits is $1$ and the Bhattacharyya parameter is 0. Block shortening means shortening the encoded $N-\bar{P}+1,\cdots,N$ bits from the last bit.

II-C Definitions of PO under Block Rate Matching

In this subsection, we define the POs for the BEC and the BMSC under block rate matching.

Firstly, we provide an equivalent description of any block puncturing and shortening:

Definition II.1

Given an odd integer $P$ and $m\in\mathbb{N}$ , s.t. $m>log_{2}P$ . For $\forall$ $N=2^{n}\geq 2^{m}$ , we define $P/2^{m}$ puncturing where we puncture $\frac{P}{2^{m}}N$ bits in the positions $\{1,\cdots,\frac{P}{2^{m}}N\}$ from a $N$ length polar code.

Definition II.2

Given an odd integer $S$ and $m\in\mathbb{N}$ , s.t. $m>log_{2}S$ . For $\forall$ $N=2^{n}\geq 2^{m}$ , we define $S/2^{m}$ shortening where we shorten $\frac{S}{2^{m}}N$ bits in the positions $\{N-\frac{S}{2^{m}}N+1,\cdots,N\}$ from a $N$ length polar code.

We provide an illustrative example explaining the similarity of different code length $N$ under $P/2^{m}$ puncturing.

Example II.1

In the case of $1/4$ puncturing, $N=4$ and $N=8$ result in the puncturing of $1$ and $2$ bits relatively. And we describe the relationship between them above.

Let $Z=(1,\epsilon,\epsilon,\epsilon)$ and $Y=(1,1,\epsilon,\epsilon,\epsilon,\epsilon,\epsilon,\epsilon)$ are the initial Bhattacharyya parameter for $N=4$ and $N=8$ relatively, where $\epsilon$ is the erasure probability of channel.Then

	$\displaystyle h(Z)$	$\displaystyle\overset{\triangle}{=}(h(Z)_{1},h(Z)_{2},h(Z)_{3},h(Z)_{4})$		(4)
		$\displaystyle=(1,2\epsilon-\epsilon^{2},\epsilon+\epsilon^{2}-\epsilon^{3},\epsilon^{3}).$		(4)

And $h(Y)$ can be derived from $h(Z)$ :

$\displaystyle h(Y)=$	$\displaystyle(2h(Z)_{1}-h^{2}(Z)_{1},h^{2}(Z)_{1},$	(5)
	$\displaystyle 2h(Z)_{2}-h^{2}(Z)_{2},h^{2}(Z)_{2},$
	$\displaystyle 2h(Z)_{3}-h^{2}(Z)_{3},h^{2}(Z)_{3},$
	$\displaystyle 2h(Z)_{4}-h^{2}(Z)_{4},h^{2}(Z)_{4}).$

Consequently, the evolution rules of the Bhattacharyya parameter in terms of the simplest fraction $P/2^{m}$ exhibit similarities. This is the reason why we define sequential puncturing in this way.

The definitions of the Bhattacharyya parameters of the synthesized channel after block rate matching are given for BEC and BMSC.

Definition II.3

For an initial BEC $W$ with polarization path $\alpha$ , define $Z_{P,m,\alpha}(x)$ be the polarized Bhattacharyya parameter under $P/2^{m}$ puncturing and $Z_{S,m,\alpha}(x)$ be the polarized Bhattacharyya parameter under $S/2^{m}$ shortening.

Definition II.4

For initial BMSC $W$ , define $Z_{P,m}(W^{\alpha})$ be the polarized Bhattacharyya parameter of channel $W^{\alpha}$ under $P/2^{m}$ puncturing and $Z_{S,m}(W^{\alpha})$ be the polarized Bhattacharyya parameter of channel $W^{\alpha}$ under $S/2^{m}$ shortening.

Then, we establish the POs based on the Bhattacharyya parameter.

Definition II.5

We write $\alpha\preceq_{P,m,BEC}\gamma$ iff

Z_{P,m,\gamma}(x)\leq Z_{P,m,\alpha}(x),\;\;\forall x\in[0,1].

(6)

We write $\alpha\preceq_{P,m,BMSC}\gamma$ , iff

Z_{P,m}(W^{\gamma})\leq Z_{P,m}(W^{\alpha}),\;\;\forall x\in[0,1].

(7)

The definitions of $\alpha\preceq_{S,m,BEC}\gamma$ and $\alpha\preceq_{S,m,BMSC}\gamma$ under shortening are the same as puncturing.

For $\alpha=\alpha_{1}\cdots\alpha_{t}$ , $Z_{P,m,\alpha}(x)$ can be calculated by $t$ iterations of either $\bar{*}$ or $\underline{*}$ convolutions, as illustrated in Fig. 1. When the convolution layer $n$ exceeds $m$ which means $m<n\leq t$ , the evolution can be simplified to the traditional case instead of (2):

\left\{\begin{aligned} &Z_{n}=2Z_{n-1}-Z_{n-1}^{2},\ \alpha_{n}=0\\ &Z_{n}=Z_{n-1}^{2},\ \alpha_{n}=1\\ \end{aligned}\right.

(8)

where $Z_{n}$ denotes the Bhattacharyya parameter under $n$ -th layer of $\alpha$ . Specially, $Z_{0}$ denotes the initial Bhattacharyya parameter of $\alpha$ and $Z_{t}=Z_{P,m,\alpha}(x)$ .

III Inherited Partial orders in the BEC under block rate matching

We give the inherited POs under the BEC in two parts. In the first part, we get some POs similar to those under the mother polar code. And in the second part, utilizing a sufficient condition, we deduce two conclusions.

III-A Some Conclusions Similar to the Old POs for the Mother Polar Codes

Firstly, we give some recursive rules of the POs in the BEC under block rate matching:

Proposition III.1

Consider the $P/2^{m}$ puncturing and $S/2^{m}$ shortening, and given strings $a$ , $b$ satisfied $|a|\geq m$ , $|b|\geq m$ , for any strings $c$ , $d$ ,

(i) if $a\preceq_{P,m,BEC}b$ , then $ac\preceq_{P,m,BEC}bc$ ;

(ii) if $a\preceq_{P,m,BEC}b$ and $c\preceq_{BEC}d$ , then $ac\preceq_{P,m,BEC}bd$ .

This is also true for $\preceq_{S,m,BEC}$ .

Proof: (i) is because $f_{c}(x)$ is monotonically increasing; use (i) and $Z_{P,m,b}(x)\in[0,1]$ , then we get (ii). $\blacksquare$

Then we deduce the relationship between $10$ and $01$ . It is then proved that under the BEC, inserting arbitrarily identical sequences before, after, and in the middle of them still maintains the PO.

Proposition III.2

$01\preceq_{P,2,BMSC}10$ and $01\preceq_{S,2,BMSC}10$ for $P=S=1$ .

Proof: From the upper bound and lower bound, we complete the proof by $\sqrt{2x^{2}-x^{4}}\geq x+x^{2}-x^{3}$ , and $\sqrt{2x^{4}-x^{6}}\geq x^{2}$ , $\forall x\in[0,1]$ . $\blacksquare$

The PO for the front insertion sequence is given below.

Lemma III.1

For any $p^{n}\in\{0,1\}^{n}$ and $n\geq 1$ , the following equality holds for any block puncture (also for block shortening):

p^{n}01\preceq_{P,m,BEC}p^{n}10

(9)

Proof: Let $\bar{P}=\frac{P}{2^{m}}N$ , $N=2^{n+2}$ and denote the binary expansion of $\bar{P}$ by $\alpha_{n+1}\cdots\alpha_{1}\alpha_{0}$ . Firstly, we can find the law: $p^{n}01$ and $p^{n}10$ are obtained from four forms after polarizing $n$ times according to $\alpha_{1}\alpha_{0}$ :

		$\displaystyle(x,x,x,x),\alpha_{1}\alpha_{0}=00$		(10)
		$\displaystyle(x,x,x,y),\alpha_{1}\alpha_{0}=01$
		$\displaystyle(x,x,y,y),\alpha_{1}\alpha_{0}=10$
		$\displaystyle(x,y,y,y),\alpha_{1}\alpha_{0}=11$

where $0<y<x<1$ . The case $(x,x,x,x)$ is obvious from the traditional polarization and the case $(x,y,y,y)$ is similar to $(x,x,x,y)$ .

1.

$(x,x,x,y)$ : $Z_{P,m,p^{n}01}=(2x-x^{2})(x+y-xy)$ , $Z_{P,m,p^{n}10}=x^{2}+xy-x^{3}y$ . Then $(2x-x^{2})(x+y-xy)\geq x^{2}+xy-x^{3}y$ holds for $0<y<x<1$ because $x(1-y)+y(1-x)\geq 0$ for $0<y<x<1$ .
2.

$(x,x,y,y)$ : $Z_{P,m,p^{n}01}=(x+y-xy)^{2}$ , $Z_{P,m,p^{n}10}=2xy-x^{2}y^{2}$ . Then $(x+y-xy)^{2}\geq 2xy-x^{2}y^{2}$ holds for $0<y<x<1$ because $x^{2}(1-y)^{2}+y^{2}(1-x)^{2}\geq 0$ for $0<y<x<1$ .

Hence, $p^{n}01\preceq_{P,m,BEC}p^{n}10$ . $\blacksquare$

The PO for the back insertion sequence is given below.

Lemma III.2

If $\alpha\preceq_{P,m,BMSC}\gamma$ holds for any block puncture, $|\alpha|=|\gamma|=n$ , then $\alpha q^{t}\preceq_{P,m,BMSC}\gamma q^{t}$ holds for any block puncture and any $q^{t}\in\{0,1\}^{t}$ . This is also true for shortening.

Proof: Claim: During the $n+t$ times polarization, any pair $(x,y)$ in the symmetric position after $n$ times polarization by $\alpha$ and $\gamma$ , is equal to the pair $(Z_{P,m,\alpha}(x),Z_{P,m,\gamma}(x))$ for some sequential puncture.

Proof of claim: actually, we can trace the initial Bhattacharyya parameter of the $2^{t}$ symmetric values:

For $i$ -th symmetric values, $1\leq i\leq 2^{t}$ : the initial Bhattacharyya parameter of the block puncture is $(z_{i},z_{i+2^{t}},\cdots,z_{i+(2^{n}-1)2^{t}})$ , where $(z_{1},\cdots,z_{2^{n+t}})$ is the initial Bhattacharyya parameter of $\alpha q^{t}$ and $\gamma q^{t}$ .

Because $\alpha\preceq_{P,m,BEC}\gamma$ holds for any block puncture, from the claim, we know $\alpha q^{t}$ and $\gamma q^{t}$ can be regarded as the result of $t$ times polarization, where each value of the $2^{t}$ positions in $\alpha q^{t}$ is greater than the symmetric value in $\gamma q^{t}$ . Consequently, after the back $t$ times polarization, $Z_{P,m,\alpha q^{t}}(x)\geq Z_{P,m,\gamma q^{t}}(x)$ . $\blacksquare$

Corollary III.1

For any $p^{n}\in\{0,1\}^{n}$ , $q^{t}\in\{0,1\}^{t}$ and $n,t\geq 1$ , the following equality holds for any block puncture (also for block shortening):

p^{n}01q^{t}\preceq_{P,m,BEC}p^{n}10q^{t}.

(11)

Then combining Lemma III.1 and Corollary III.1, we can obtain Theorem III.1 by induction on $\ell$ like [20].

Theorem III.1

For any $n,t,\ell\geq 0$ $p^{n}\in\{0,1\}^{n}$ , $q^{t}\in\{0,1\}^{t}$ , $r^{\ell}\in\{0,1\}^{\ell}$ , we have

p^{n}0r^{\ell}1q^{t}\preceq_{P,m,BEC}p^{n}1r^{\ell}0q^{t}

(12)

This is also true for $\preceq_{S,m,BEC}$ .

III-B A Sufficient Condition for Inherited POs in BEC

We firstly obtain a sufficient condition for inherited POs in the BEC above, and other conclusions in this subsection are obtained by verifying the sufficient condition.

Lemma III.3

Consider the $P/2^{m}$ puncturing or $S/2^{m}$ shortening, and given $\tau_{1},\tau_{2}\in\{0,1\}^{m}$ . If

h_{\tau_{i}}:=f_{\tau_{i}}^{-1}\circ Z_{\tau_{i}},\;\;i=1,2,

(13)

satisfy that $h_{\tau_{1}}(x)\leq h_{\tau_{2}}(x),\forall x\in[0,1]$ , then

\tau_{2}\alpha\preceq_{BEC}\tau_{1}\gamma\Rightarrow\tau_{2}\alpha\preceq_{P,m,BEC}\tau_{1}\gamma.

(14)

where $f_{\tau_{i}}$ is the traditional Bhattacharyya parameter defined by (1), and $Z_{\tau_{i}}$ is the rate-matched Bhattacharyya parameter defined by (2). $\preceq_{BEC}$ is the traditional PO under the BEC [21].

Proof: $Z_{\tau_{2}\alpha}(x)=f_{\alpha}\circ f_{\tau_{2}}(h_{\tau_{2}}(x))\geq f_{\alpha}\circ f_{\tau_{2}}(h_{\tau_{1}}(x))\geq f_{\gamma}\circ f_{\tau_{1}}(h_{\tau_{1}}(x))=Z_{\tau_{1}\gamma}(x).$ $\blacksquare$

When $P=1$ , we get a inherited PO related to a form of $0^{p}1^{q}$ , $p+q=m$ .

Theorem III.2

For any $1/2^{m}$ puncturing,

		$\displaystyle 0^{p}1^{m-p}\alpha\preceq_{BEC}0^{q}1^{m-q}\gamma$		(15)
	$\displaystyle\Rightarrow$	$\displaystyle 0^{p}1^{m-p}\alpha\preceq_{1,m,BEC}0^{q}1^{m-q}\gamma,\forall 0\leq q\leq p\leq m.$		(15)

Proof:

\Leftarrow h_{0^{m-k-1}1^{k+1}}(x)\leq h_{0^{m-k}1^{k}}(x),\forall x\in[0,1],m>k\in\mathbb{N}^{+}

(16)

		$\displaystyle\Leftrightarrow\forall x\in[0,1],\left(1-(1-(1-x)^{2^{m-k-1}})^{1-2^{-k-1}}\right)^{2}$		(17)
		$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\left(1-(1-(1-x)^{2^{m-k}})^{1-2^{-k}}\right)\geq 0.$		(17)

$\overset{t=(1-x)^{2^{m-k-1}}}{\Longleftrightarrow}$

\left(1-(1-t)^{1-2^{-k-1}}\right)^{2}-\left(1-(1-t^{2})^{1-2^{-k}}\right)\geq 0,\;\;\forall t\in[0,1].

(18)

$\Leftrightarrow$

(1-t)^{2-2^{-k}}-2(1-t)^{1-2^{-k-1}}+(1-t^{2})^{1-2^{-k}}\geq 0,\;\;\forall t\in[0,1].

(19)

$\Leftrightarrow$

(1-t)^{1-2^{-k-1}}-2+(1+t)^{1-2^{-k}}(1-t)^{2^{-k-1}}\geq 0,\;\;\forall t\in[0,1].

(20)

$\overset{u=1-t}{\Longleftrightarrow}$

u+(2-u)^{1-2^{-k}}\geq 2u^{2^{-k-1}},\;\;\forall u\in[0,1].

(21)

Denote $f(u)=u+(2-u)^{1-2^{-k}}-2u^{2^{-k-1}}$ , then $f(1)=0,f^{\prime}(1)=0,$

	$\displaystyle f^{\prime\prime}(u)$	$\displaystyle=(1-2^{-k})2^{-k}(2-u)^{-2^{-k}-1}$		(22)
		$\displaystyle+2^{-k}(1+2^{-k-1})u^{-2^{-k-1}-2}\geq 0$		(22)

thus $f(u)\geq 0,\;\;\forall x\in[0,1]$ , then (16) is proved. $\blacksquare$

Here is a conclusion for any block rate matching.

Theorem III.3

Consider the $P/2^{m}$ puncturing or $S/2^{m}$ shortening, and $|\gamma|=m-1$ , we have

\gamma 0\alpha\preceq_{BEC}\gamma 1\beta\Rightarrow\gamma 0\alpha\preceq_{P,m,BEC}\gamma 1\beta

(23)

Proof: we only need to proof $h_{\gamma 0}(x)\geq h_{\gamma 1}(x),\forall x\in[0,1]$ .

f_{\gamma 1}^{-1}(x)=f_{\gamma}^{-1}(\sqrt{x}),f_{\gamma 0}^{-1}(x)=f_{\gamma}^{-1}(1-\sqrt{1-x}).

(24)

Let

		$\displaystyle Z_{\gamma 1}(x)=U_{\gamma}(x)L_{\gamma}(x),$		(25)
		$\displaystyle Z_{\gamma 0}(x)=U_{\gamma}(x)+L_{\gamma}(x)-U_{\gamma}(x)L_{\gamma}(x).$		(25)

Then

\Leftarrow f_{\gamma}^{-1}(1-\sqrt{1-Z_{\gamma 0}(x)})\geq f_{\gamma}^{-1}(\sqrt{Z_{\gamma 1}(x)})

(26)

\Leftarrow\left(Z_{\gamma 0}(x)+Z_{\gamma 1}(x)\right)^{2}\geq 4Z_{\gamma 1}(x)

(27)

\Leftarrow(U_{\gamma}(x)-L_{\gamma}(x))^{2}\geq 0.

(28)

$\blacksquare$

IV New Partial Orders in BMSC

In this section, we establish a general PO from BEC to BMSC under block rate matching. While the process of proof is different from that in the mother polar code. Firstly, we construct a convolution mapping in Lemma IV.1. Then we use this mapping to prove that geometric mean pair exhibits a superior polarization effect in Proposition IV.2. Furthermore, we can proof a critical inequality in Lemma IV.2 by Proposition IV.2. Finally, utilizing the inequality and the technology of the upper and lower bounds like [21], we establish the general PO from BEC to BMSC under block rate matching.

Before we construct the convolution mapping, we see what kind of two positions do convolution for defining the convolution mapping.

Proposition IV.1

Let $N=2^{n}$ , $i,j\in\{1,2,\cdots,N\},2^{s}<i\leq 2^{s+1},2^{q}<j\leq 2^{q+1}$ , $i<j$ then $i$ and $j$ do convolution in some layer iff

1.

$j-i=2^{q}$ , if $s<q$
2.

$i-2^{q}$ and $j-2^{q}$ do convolution in some layer, if $s=q$

Then we give the definition of convolution mapping before we construct the convolution mapping between two consecutive integer sets.

Definition IV.1

A one-to-one mapping $f:\mathcal{X}\rightarrow\mathcal{Y}$ is called a convolution mapping if for $\forall x\in\mathcal{X}$ , $x$ and $f(x)$ do convolution in some layer.

The convolution mapping between two consecutive integer sets is constructed as follows.

Lemma IV.1

$\forall\ K\in N^{+}$ ,define $\mathcal{X}=\{1,2,\cdots,K\},\mathcal{Y}=\{K+1,K+2,\cdots,2K\}$ ,there exist a convolution mapping $f:\mathcal{X}\rightarrow\mathcal{Y}$ .

Proof: See Appendix -A. $\blacksquare$

An example is given to facilitate our view of the convolution mapping in the form.

Example IV.1

$\mathcal{X}=\{1,2,3,4,5\}$ , $\mathcal{Y}=\{6,7,8,9,10\}$ , then

f(x)=\left\{\begin{aligned} &x+2^{3},when\ x\in\{1,2\}\\ &x+2^{2},when\ x\in\{3,4\}\\ &x+2^{0},when\ x=5\\ \end{aligned}\right.

(29)

Remark IV.1

The parameter $k$ in the proof of Lemma IV.1 represents the polarization layer in the evolution of Bhattacharyya parameters. And the pair $(x,f(x))$ denotes the position corresponding to different values of Bhattacharyya parameters.

Next we present a crucial polarization rule by utilizing the convolution mapping. It reveals that the more uniform the initial Bhattacharyya parameters are, the smaller the the polarized Bhattacharyya parameters are.

Proposition IV.2

For initial Bhattacharyya parameters $Z=(\underbrace{a,\cdots,a}_{P},b,\cdots,b)$ , where $a,b\in(0,1)$ , $|Z|=N$ , $1\leq P\leq N$ , $Z_{k}$ and $Z_{k+1}$ are defined as follows. $Z_{k}$ have $k$ positions $\sqrt{ab}$ in $a$ positions and $k$ positions $\sqrt{ab}$ in $b$ positions of $Z$ . And for $i,j\in\{1,\cdots,N\}$ , if $z_{i}=a$ and $z_{j}=b$ do convolution in the outermost layer among all the $\sqrt{ab}$ pairs, then we replace $(z_{i},z_{j})$ by $(\hat{z}_{i},\hat{z}_{j})=(\sqrt{ab},\sqrt{ab})$ in $Z_{k}$ denoted by $Z_{k+1}$ . Then we have $h(Z_{k+1})\leq h(Z_{k})$ .

Proof: See Appendix -B. $\blacksquare$

Constructing the convolution mapping to prove Proposition IV.2 is to obtain the following inequality associated with Bhattacharyya parameters.

Lemma IV.2

For $P/2^{m}$ puncturing and $S/2^{m}$ shortening,we have

Z_{P,m,\beta}(x^{2})\geq Z_{P,m,1\beta}(x),\forall\mid\beta\mid=m

(30)

Proof: See Appendix -C. $\blacksquare$

Here is an illustrative example about Lemma IV.2 for understanding.

Example IV.2

Consider $1/4$ puncturing polar code, $\mid\alpha\mid=2$ :

Z_{P,2,\alpha}(x^{2})\geq Z_{P,2,1\alpha}(x)

(31)

where

Z_{P,2,\alpha}(x^{2})=\left\{\begin{aligned} &1,\alpha=00\\ &2x^{2}-x^{4},\alpha=01\\ &x^{2}+x^{4}-x^{6},\alpha=10\\ &x^{6},\alpha=11\\ \end{aligned}\right.

(32)

Z_{P,2,1\alpha}(x)=\left\{\begin{aligned} &1-(1-(x+x^{2}-x^{3}))^{2},\alpha=00\\ &(x+x^{2}-x^{3})^{2},\alpha=01\\ &2x^{3}-x^{6},\alpha=10\\ &x^{6},\alpha=11\\ \end{aligned}\right.

(33)

The final step of preparation is to analyze the upper and lower bounds of Bhattacharyya parameters as discussed in [21].

Lemma IV.3

Given BMSC $W$ with $Z(W)=x$ , then for $P/2^{m}$ puncturing and $\mid\alpha\mid=m$ ,

\sqrt{Z_{P,m,\alpha}(x^{2})}\leq Z_{P,m}(W^{\alpha})\leq Z_{P,m,\alpha}(x)

(34)

For $\tau=\alpha\gamma$ ,

\sqrt{f_{\gamma}\circ Z_{P,m,\alpha}(x^{2})}\leq Z_{P,m}(W^{\tau})\leq f_{\gamma}\circ Z_{P,m,\alpha}(x)

(35)

This is also true for $S/2^{m}$ shortening.

Proof: We proof for puncturing as an example by induction. Firstly, when $|\alpha|=1$ , it is true obviously.

$\alpha=\gamma 0$ :

$\displaystyle Z_{P,m}(W^{\alpha})$	$\displaystyle\overset{(a)}{\geq}\sqrt{2Z^{2}_{P,m}(W^{\gamma})-Z^{4}_{P,m}(W^{\gamma})}$	(36)
	$\displaystyle\overset{(b)}{\geq}\sqrt{2Z_{P,m,\gamma}(x^{2})-Z^{2}_{P,m,\gamma}(x^{2})}$	(37)
	$\displaystyle=\sqrt{Z_{P,m,\alpha}(x^{2})}.$	(38)

$\displaystyle Z_{P,m}(W^{\alpha})$	$\displaystyle\overset{(c)}{\leq}2Z_{P,m}(W^{\gamma})-Z_{P,m}^{2}(W^{\gamma})$	(39)
	$\displaystyle\overset{(d)}{\leq}2Z_{P,m,\gamma}(x)-Z^{2}_{P,m,\gamma}(x)$	(40)
	$\displaystyle=Z_{P,m,\alpha}(x).$	(41)

2.

It is obviously for the case $\alpha=\gamma 1$ because $Z_{P,m}(W^{\alpha})=Z_{P,m}^{2}(W^{\gamma})$ .

where $(a)$ and $(c)$ are from the lower and upper bounds [20], $(b)$ and $(d)$ are from the induction.

$\blacksquare$

Leveraging Lemma IV.2 and Lemma IV.3, we can derive our main theorem, which deriving the PO of the BMSC by leveraging the PO of the BEC under block rate matching.

Theorem IV.1

For block puncturing, $\mid\gamma\mid=\mid\alpha\mid\geq m$ , we have

\gamma\preceq_{P,m,BEC}\alpha\Rightarrow\gamma 1\preceq_{P,m,BMSC}1\alpha.

(42)

This is also true for $S/2^{m}$ shortening.

Proof: According to Lemma IV.3

	$\displaystyle Z_{P,m,\alpha}(x^{2})$	$\displaystyle\leq f_{1}\circ f_{1}^{-1}\circ Z_{P,m,\gamma}\circ f_{1}(x)$		(43)
		$\displaystyle\leq f_{1}\circ Z_{P,m}(W^{\gamma})=Z_{P,m}(W^{\gamma 1})$		(43)

And use Lemma IV.2, we have

Z_{P,m}(W^{1\alpha})\leq Z_{P,m,1\alpha}(x)\leq Z_{P,m,\alpha}(x^{2})\leq Z_{P,m}(W^{\gamma 1})

(44)

So $\gamma 1\preceq_{P,m,BMSC}1\alpha$ . $\blacksquare$

The following proposition is a corollary of Theorem IV.1.

Proposition IV.3

For block puncturing, $\mid\gamma\mid=\mid\alpha\mid\geq m$ , we have

1\gamma\preceq_{P,m,BEC}\alpha 1\Rightarrow\gamma\preceq_{P,m,BMSC}\alpha.

(45)

This is also true for $S/2^{m}$ shortening.

Proof: $1\gamma\preceq_{P,m,BEC}\alpha 1\Rightarrow Z_{\alpha 1}(x)\leq Z_{1\gamma}(x),x\in[0,1]\Rightarrow Z_{\alpha}(x)\leq\sqrt{Z_{1\gamma}(x)},x\in[0,1]\Rightarrow Z(W^{\alpha})\leq Z_{\alpha}(x)\leq\sqrt{Z_{1\gamma}(x)}\leq\sqrt{Z_{\gamma}(x^{2})}\leq Z(W^{\gamma})\Rightarrow\gamma\preceq_{P,m,BMSC}\alpha$ . $\blacksquare$

V Simulation

When $N=1024$ and considering $1/4$ block puncturing, there are $C_{768}^{2}=294528$ path pairs in total. According to Theorem IV.1, we find 198258 pairs satisfy the PO $\preceq_{1,2,BMSC}$ . If the length $n_{0}$ of the leading identical sequence components exceed $2$ , the partial order of the two sequences match that of traditional polar codes after removing the first $n_{0}$ bits. So in this case, we refer to [21] to check the pair. By employing this method, we identify 212226 pairs.

We generate the information set $\mathcal{A}_{GA}$ by GA reconstruction under rate matching [2] at $SNR=2.2dB$ , and we observe that $\mathcal{A}_{GA}$ follows the partial order among all the 212226 pairs. It verifies the POs from Theorem IV.1 are beneficial for constructing block punctured polar codes.

Then we generate the information set $\mathcal{A}_{PW}$ by PW reliability sequence in [3]. And $\mathcal{A}_{improved}$ is generated by replace the positions in $\mathcal{A}_{PW}$ utilizing the PO pairs from IV.1, which are contrary to PW sequence. Fig. 2 presents a performance comparison between the two polar codes under block puncturing and shortening. It is observed that $\mathcal{A}_{improved}$ has a gain of $0.13dB$ compared to $\mathcal{A}_{PW}$ under block puncturing. This illustrates that the PW construction can be further optimized from the perspective of PO.

VI Conclusion

In this paper, we firstly establish partial orders under block rate matching. And we introduce a sufficient condition for verifying the inherited POs of BEC under block rate matching. For the research in the BMSC, we demonstrate the property that the geometric mean of a pair of Bhattacharyya parameters decreases after polarization. By combining this result with the technique of upper and lower bounds of Bhattacharyya parameters, we establish that under block rate matching, the POs of BMSC can be derived from the POs of BEC. Finally, we verify that our work has guiding significance for the construction of polar codes under block puncturing.

References

[1] E. Arikan. Channel polarization: A method for constructing capacity-achieving codes. In 2008 IEEE International Symposium on Information Theory, pages 1173–1177, 2008.
[2] P. Trifonov. Efficient design and decoding of polar codes. In IEEE Transactions on Communications, 60(11):3221-3227, 2012.
[3] G. He, J.-C. Belfiore, I. Land, G.-H. Yang, X.-C. Liu, Y. Chen, R. Li, J. Wang, Y.-Q. Ge, R. Zhang, and W. Tong. Beta-expansion: A theoretical framework for fast and recursive construction of polar codes. In 2017 IEEE Global Communications Conference, pages 1–6, 2017.
[4] 3GPP, ”NR; Multiplexing and channel coding”, 3GPP TS 38.212,15.5.0, Mar. 2019.
[5] Roth, Ron. Introduction to coding theory. Springer-Verlag 1999.
[6] F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. Amsterdam: North-Holland 1977.
[7] Boaz Shuval and Ido Tal. Strong polarization for shortened and punctured polar codes. In arXiv:2401.16833v1, 2024.
[8] Niu, Kai and Chen, Kai and Lin, Jia-Ru. Beyond turbo codes: Rate-compatible punctured polar codes. In 2013 IEEE International Conference on Communications (ICC), 2013.
[9] Wang, Runxin and Liu, Rongke. A novel puncturing scheme for polar codes. In IEEE Communications Letters, 2014.
[10] Bioglio, Valerio and Gabry, Frederic and Land, Ingmar. Low-complexity puncturing and shortening of polar codes. In 2017 IEEE Wireless Communications and Networking Conference Workshops (WCNCW), 2017.
[11] Oliveira, Robert M. and de Lamare, Rodrigo C. Rate-compatible polar codes based on polarization-driven shortening. In IEEE Communications Letters, 2018.
[12] Oliveira, Robert M. and de Lamare, Rodrigo C. Puncturing based on polarization for polar codes in 5G networks. In 2018 15th International Symposium on Wireless Communication Systems (ISWCS), 2018.
[13] Yao, Xinyuanmeng and Ma, Xiao. A balanced tree approach to construction of length-flexible polar codes. In IEEE Transactions on Communications, 2024.
[14] Tonnellier, Thibaud and Cavatassi, Adam and Gross, Warren J. Length-compatible polar codes: A survey : (invited paper). In 2019 53rd Annual Conference on Information Sciences and Systems (CISS), 2019.
[15] T. -C. Lin and H. -P. Wang. Optimal self-dual inequalities to order polarized BECs. In IEEE International Symposium on Information Theory (ISIT), 2023.
[16] H. -P. Wang and V. -F. Drăgoi. Fast methods for ranking synthetic BECs. In IEEE International Symposium on Information Theory (ISIT), 2023.
[17] H. -P. Wang and C. -W. Chin. Density devolution for ordering synthetic channels. In IEEE International Symposium on Information Theory (ISIT), 2023.
[18] I Tal, A Vardy. How to construct polar codes. In IEEE Transactions on Information Theory, 2011.
[19] C Schurch. A partial order for the synthesized channels of a polar code. In IEEE International Symposium on Information Theory (ISIT), 2016.
[20] Wu W , Siegel P H. Generalized partial orders for polar code bit-channels. In IEEE Transactions on Information Theory, 2019.
[21] Liuquan Yao, Zhichao Liu, Yuan Li, Huazi Zhang, Jun Wang, Guiying Yan,and Zhi-Ming Ma. New partial orders of polar codes for BMSC. In IEEE International Symposium on Information Theory (ISIT), pp. 2192-2197, 2024.
[22] I. Tal and A. Vardy. List decoding of polar codes. In IEEE Transactions on Information Theory, vol. 61, no. 5, pp. 2213-2226, May 2015.
[23] LK. Niu and K. Chen. CRC-aided decoding of polar codes. In IEEE Communications Letters, vol. 16, no. 10, pp. 1668-1671, October 2012.

-A Proof of Lemma IV.1

Let $2^{q}<K\leq 2^{q+1}$ . In order to establish the one-to-one mapping $f$ , we firstly divide $\mathcal{X},\mathcal{Y}$ into several sets $\mathcal{X}_{k}=\{x_{k}^{1},\cdots,x_{k}^{t(k)}\},\mathcal{Y}_{k}=\{y_{k}^{1},\cdots,y_{k}^{t(k)}\}$ :

1. $\mathcal{X}_{1}=\{1,2\cdots,2K-2^{q+1}\}$ , $\mathcal{Y}_{1}=\{1+2^{q+1},\cdots,2K\}$ , $2^{q+1}\leq y_{1}^{t(k)}-x_{1}^{1}=2K-1<2^{q+2}$ .

2. $\mathcal{X}_{k}=\{x_{k-1}^{t(k-1)}+1,\cdots,y_{k-1}^{t(k-1)}-1-2^{m_{k}}\}$ , $\mathcal{Y}_{k}=\{x_{k-1}^{t(k-1)}+1+2^{m_{k}},\cdots,y_{k-1}^{t(k-1)}-1\}$ , $x_{k}^{1}\leq x_{k}^{t(k)}<y_{k}^{1}\Leftrightarrow 2^{m_{k}}\leq y_{k}^{t(k)}-x_{k}^{1}<2^{m_{k}+1}$ until $y_{k}^{1}-x_{k}^{t(k)}=1\Leftrightarrow y_{k}^{t(k)}-x_{k}^{1}=2^{m_{k}+1}-1,$ denoted by $k_{0}$ .

Then $\mathcal{X}=\mathop{\cup}_{k=1}^{k_{0}}\mathcal{X}_{k},\mathcal{Y}=\mathop{\cup}_{k=1}^{k_{0}}\mathcal{Y}_{k}$ ,and we construct $f$ by $f(x_{k}^{i})=y_{k}^{i},1\leq k\leq k_{0},1\leq i\leq t(k)$ . $f$ is a convolution mapping because from Proposition IV.1: $2^{q+1}+\mathop{\sum}_{k=1}^{k_{0}}(-1)^{k}2^{m_{k}}+1$ and $2^{q+1}+\mathop{\sum}_{k=1}^{k_{0}+1}(-1)^{k}2^{m_{k}}+1$ do convolution iff $1$ and $2^{m_{k_{0}}}+1$ do convolution; $2^{q+1}+\mathop{\sum}_{k=1}^{k_{0}}(-1)^{k}2^{m_{k}}$ and $2^{q+1}+\mathop{\sum}_{k=1}^{k_{0}+1}(-1)^{k}2^{m_{k}}$ do convolution iff $2^{m_{k_{0}}}$ and $2^{m_{k_{0}}+1}$ do convolution. $\blacksquare$

-B Proof of Proposition IV.2

We induct on code length $N$ : assume it is right for $\frac{N}{2}$ .

(i) $i\leq\frac{N}{2},j>\frac{N}{2}(or\ i>\frac{N}{2},j\leq\frac{N}{2})$ : then $j=i+\frac{N}{2}$

z_{i}\bar{*}z_{j}\geq 2\sqrt{z_{i}z_{j}}-z_{i}z_{j};z_{i}z_{j}=z_{i}z_{j}

(46)

it means $h(Z_{k+1})\leq h(Z_{k})$ .

(ii) $i\leq\frac{N}{2},j\leq\frac{N}{2}(or\ i>\frac{N}{2},j>\frac{N}{2})$ : then $P\neq\frac{N}{2}$

		$\displaystyle z_{i}\bar{*}z_{i+\frac{N}{2}}=a+b-ab;$		(47)
		$\displaystyle z_{j}\bar{*}z_{j+\frac{N}{2}}=2b-b^{2};$		(47)

\displaystyle z_{i}\underline{*}z_{i+\frac{N}{2}}=ab;z_{j}\underline{*}z_{j+\frac{N}{2}}=b^{2}

(48)

		$\displaystyle\hat{z}_{i}\bar{*}z_{i+\frac{N}{2}}=\sqrt{ab}+b-\sqrt{ab}b;$		(49)
		$\displaystyle\hat{z}_{j}\bar{*}z_{j+\frac{N}{2}}=\sqrt{ab}+b-\sqrt{ab}b;$		(49)

\displaystyle\hat{z}_{i}\underline{*}z_{i+\frac{N}{2}}=\hat{z}_{j}\underline{*}z_{j+\frac{N}{2}}=\sqrt{ab}b=\sqrt{z_{i}z_{i+\frac{N}{2}}}\underline{*}\sqrt{z_{j}z_{j+\frac{N}{2}}}

(50)

Because for $i,j\leq\frac{N}{2}$ , $z_{i}$ and $z_{j}$ can not convolve in the outermost layer, none of the $\sqrt{ab}$ pairs can convolve in the outermost layer with the condition of lemma. Let $z_{up}$ and $z_{down}$ denote values in the front and back of $Z_{k}$ relatively, then we have the following intuition

\left\{\begin{aligned} &z_{up}\in\{a,b,\sqrt{ab}\},z_{down}=b,when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2}\\ &z_{up}\in\{a\},z_{down}\in\{a,b,\sqrt{ab}\},when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2}\\ \end{aligned}\right.

(51)

On the one hand,

\left\{\begin{aligned} &Z^{(1)}_{back}=(ab,\cdots,ab,b^{2},\cdots,b^{2}),when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2}\\ &Z^{(1)}_{back}=(a^{2},\cdots,a^{2},ab,\cdots,ab),when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2}\\ \end{aligned}\right.

(52)

There are $k$ positions containing $\sqrt{ab}b\ (or\ \sqrt{ab}a)$ instead of $ab$ and $k$ positions containing $\sqrt{ab}b\ (or\ \sqrt{ab}a)$ instead of $b^{2}\ (or\ a^{2})$ in $Z^{(1)}_{back}$ .

And $\hat{Z}^{(1)}_{back}$ is generated by replacing a pair of $Z^{(1)}_{back}$ as follows:

\left\{\begin{aligned} \hat{Z}^{(1)}_{back}:&replace\ (z^{(1)}_{i},z^{(1)}_{j})\ by\ (\sqrt{ab}b,\sqrt{ab}b),\\ &when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2}\\ \hat{Z}^{(1)}_{back}:&replace\ (z^{(1)}_{i-\frac{N}{2}},z^{(1)}_{j-\frac{N}{2}})\ by\ (\sqrt{ab}a,\sqrt{ab}a),\\ &when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2}\\ \end{aligned}\right.

(53)

We know that all the averaged pairs either belong to the front half or the back half, so $(z^{(1)}_{i},z^{(1)}_{j})$ or $(z^{(1)}_{i-\frac{N}{2}},z^{(1)}_{j-\frac{N}{2}})$ remains in the outermost layer among all the averaged pairs. By induction,we have

h(\hat{Z}^{(1)}_{back})\leq h(Z^{(1)}_{back})

(54)

On the other hand,

\left\{\begin{aligned} Z^{(1)}_{front}=&(a+b-ab,\cdots,a+b-ab,2b-b^{2},\cdots,2b-b^{2}),\\ &when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2}\\ Z^{(1)}_{front}=&(2a-a^{2},\cdots,2a-a^{2},a+b-ab,\cdots,a+b-ab),\\ &when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2}\\ \end{aligned}\right.

(55)

There are $k$ positions containing $\sqrt{ab}+b-\sqrt{ab}b$ (or $\sqrt{ab}+a-\sqrt{ab}a$ ) instead of $a+b-ab$ and $k$ positions containing $\sqrt{ab}+b-\sqrt{ab}b$ (or $\sqrt{ab}+a-\sqrt{ab}a$ ) instead of $2b-b^{2}\ (or\ 2a-a^{2})$ in $Z^{(1)}_{front}$ .

Similarly, $\hat{Z}^{(1)}_{front}$ is generated by replacing a pair of $Z^{(1)}_{front}$ as follows:

\left\{\begin{aligned} \hat{Z}^{(1)}_{front}:&when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2},replace\ (z^{(1)}_{i},z^{(1)}_{j})\\ &by\ (\sqrt{ab}+b-\sqrt{ab}b,\sqrt{ab}+b-\sqrt{ab}b)\\ \hat{Z}^{(1)}_{front}:&when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2},replace\ (z^{(1)}_{i-\frac{N}{2}},z^{(1)}_{j-\frac{N}{2}})\\ &by\ (\sqrt{ab}+a-\sqrt{ab}a,\sqrt{ab}+a-\sqrt{ab}a)\\ \end{aligned}\right.

(56)

As a medium step, we generate $\bar{Z}_{front}^{(1)}$ by replacing the same pair of $Z^{(1)}_{front}$ as follows:

\left\{\begin{aligned} \bar{Z}^{(1)}_{front}:&when\ P<\frac{N}{2},i\leq\frac{N}{2},j\leq\frac{N}{2},replace\ (z^{(1)}_{i},z^{(1)}_{j})\\ &by\ (\sqrt{(a+b-ab)(2b-b^{2})},\sqrt{(a+b-ab)(2b-b^{2})})\\ \bar{Z}^{(1)}_{front}:&when\ P>\frac{N}{2},i>\frac{N}{2},j>\frac{N}{2},replace\ (z^{(1)}_{i-\frac{N}{2}},z^{(1)}_{j-\frac{N}{2}})\\ &by\ (\sqrt{(a+b-ab)(2a-a^{2})},\sqrt{(a+b-ab)(2a-a^{2})})\\ \end{aligned}\right.

(57)

According to $(z^{(1)}_{i},z^{(1)}_{j})$ or $(z^{(1)}_{i-\frac{N}{2}},z^{(1)}_{j-\frac{N}{2}})$ remains in the outermost layer among all the averaged pairs, and

\left\{\begin{aligned} &\sqrt{(z_{i}\bar{*}z_{i+\frac{N}{2}})(z_{j}\bar{*}z_{j+\frac{N}{2}})}=\sqrt{(a+b-ab)(2b-b^{2})},P<\frac{N}{2}\\ &\sqrt{(z_{i}\bar{*}z_{i-\frac{N}{2}})(z_{j}\bar{*}z_{j-\frac{N}{2}})}=\sqrt{(a+b-ab)(2a-a^{2})},P>\frac{N}{2}\\ \end{aligned}\right.

(58)

Then by the induction, we conclude that

h(\bar{Z}^{(1)}_{front})\leq h(Z^{(1)}_{front})

(59)

And use the inequality

\sqrt{ab}+b-\sqrt{ab}b\leq\sqrt{(a+b-ab)(2b-b^{2})},\forall a,b\in[0,1]

(60)

then we have

h(\hat{Z}^{(1)}_{front})\leq h(\bar{Z}^{(1)}_{front})

(61)

h(\hat{Z}^{(1)}_{front})\leq h(\bar{Z}^{(1)}_{front})\leq h(Z^{(1)}_{front})

(62)

Finally,

	$\displaystyle h(Z_{k+1})$	$\displaystyle=(h(\hat{Z}^{(1)}_{front}),h(\hat{Z}^{(1)}_{back}))$		(63)
		$\displaystyle\leq(h(Z^{(1)}_{front}),h(Z^{(1)}_{back}))=h(Z_{k})$		(63)

$\blacksquare$

-C Proof of Lemma IV.2

For simplification, denote the initial Bhattacharyya parameters of $Z_{P,m,\beta}(x^{2})$ by $Z$ , and one time down polarization of the initial Bhattacharyya parameters of $Z_{P,m,1\beta}(x)$ by $\hat{Z}$ . Then $Z$ and $\hat{Z}$ can be written as

\left\{\begin{aligned} P\leq\frac{N}{2}:&Z=(\underbrace{1,\cdots,1}_{P},\underbrace{x^{2},\cdots,x^{2}}_{P},\underbrace{x^{2},\cdots,x^{2}}_{N-2P}),\\ &\hat{Z}=(\underbrace{x,\cdots,x}_{2P},\underbrace{x^{2},\cdots,x^{2}}_{N-2P})\\ P>\frac{N}{2}:&Z=(\underbrace{1,\cdots,1}_{2P-N},\underbrace{1,\cdots,1}_{N-P},\underbrace{x^{2},\cdots,x^{2}}_{N-P}),\\ &\hat{Z}=(\underbrace{1,\cdots,1}_{2P-N},\underbrace{x,\cdots,x}_{2(N-P)})\\ \end{aligned}\right.

(64)

According to Lemma IV.1, when $P\leq\frac{N}{2}$ , the first $2P$ numbers of $Z$ can be partitioned into $P$ pairs. When $P>\frac{N}{2}$ , the symmetry between the front and back positions ensures the validity of this partitioning. It is essential to highlight that these $P$ pairs need to be arranged in ascending order of layers to satisfy the condition of Proposition IV.2.

Then we replace each $(1,x^{2})$ pair in $Z$ with $(x,x)$ in turn among the $P$ pairs. Let $Z_{k}$ represent the initial Bhattacharyya parameters with $k$ averaged pairs, where the first $k-1$ averaged pairs are identical to those of $Z_{k-1}$ .

Use Proposition IV.2 we get

h(Z_{k-1})\leq h(Z_{k}),\forall 1\leq k\leq P

(65)

h(\hat{Z}):=h(Z_{P})\leq h(Z_{P-1})\leq\cdots h(Z_{1})\leq h(Z_{0})=:h(Z)

(66)

It means

Z_{P,m,\beta}(x^{2})\geq Z_{P,m,1\beta}(x),\forall\mid\beta\mid=m

(67)

$\blacksquare$