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Partial Orders in Rate-Matched Polar Codes

Zhichao Liu, Liuquan Yao, Yuan Li, Huazi Zhang, Jun Wang, Guiying Yan, Zhiming Ma

TL;DR

The partial order (POs) are established for both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC) under any block rate-matched polar codes.

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 β-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.

Partial Orders in Rate-Matched Polar Codes

TL;DR

The partial order (POs) are established for both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC) under any block rate-matched polar codes.

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 β-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.

Paper Structure

This paper contains 16 sections, 16 theorems, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organizations and Notations
  4. Preliminaries
  5. Polar Codes
  6. Rate Matching
  7. Definitions of PO under Block Rate Matching
  8. Inherited Partial orders in the BEC under block rate matching
  9. Some Conclusions Similar to the Old POs for the Mother Polar Codes
  10. A Sufficient Condition for Inherited POs in BEC
  11. New Partial Orders in BMSC
  12. Simulation
  13. Conclusion
  14. Proof of Lemma \ref{['oto']}
  15. Proof of Proposition \ref{['avpa']}
  16. ...and 1 more sections

Key Result

Proposition 3.1

Consider the $P/2^m$ puncturing and $S/2^m$ shortening, and given strings $a$, $b$ satisfied $|a|\ge m$, $|b|\ge m$, 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

Figures (2)

  • Figure 1: $h(Z)$ as the polarized vector of $Z$ for $N=4$.
  • Figure 2: SCL performance for two different information sets $\mathcal{A}_{PW}$ and $\mathcal{A}_{improved}$ with CRC length $12$ and list $L \in \{1,2,4,8\}$.

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.1
  • Lemma 3.2
  • ...and 16 more