Construction Methods Based on Minimum Weight Distribution for Polar Codes with Successive Cancellation List Decoding

TL;DR

This paper develops a channel-independent construction framework for polar codes by leveraging minimum weight distribution (MWD). It introduces the MWD sequence, built from partial MWD to order synthetic channels under partial order, and proves it is optimal with respect to the minimum weight bound MWUB. It also couples MWD with an entropy constraint through the ECBS algorithm, which greedily swaps information and frozen bits to satisfy a list-size bound while preserving $d_{\min}$, thereby approaching MWUB for SCL decoding as $L$ grows. Empirical results show substantial gains for short codes over 5G polar designs and demonstrate that ECBS can bring performance close to the MWUB, validating the practical value of MWD-based, channel-agnostic polar-code construction. Overall, the work provides a principled, nested construction method and a scalable algorithm to enhance SCL performance without channel knowledge.

Abstract

Minimum weight distribution (MWD) is an important metric to calculate the first term of union bound called minimum weight union bound (MWUB). In this paper, we first prove the maximum likelihood (ML) performance approaches MWUB as signal-to-noise ratio (SNR) goes to infinity and provide the deviation when MWD and SNR are given. Then, we propose a nested reliability sequence, namely MWD sequence, to construct polar codes independently of channel information. In the sequence, synthetic channels are sorted by partial MWD which is used to evaluate the influence of information bit on MWD and we prove the MWD sequence is the optimum sequence evaluated by MWUB for polar codes obeying partial order. Finally, we introduce an entropy constraint to establish a relationship between list size and MWUB and propose a heuristic construction method named entropy constraint bit-swapping (ECBS) algorithm, where we initialize information set by the MWD sequence and gradually swap information bit and frozen bit to satisfy the entropy constraint. The simulation results show the MWD sequence is more suitable for constructing polar codes with short code length than the polar sequence in 5G and the ECBS algorithm can improve MWD to show better performance as list size increases.

Figures (8)

• Figure 1: $P_{\tt UB}$, $P_{\tt MWUB}$, $P_{\tt LB}$, $P_{\tt LUB}$, $\delta_1$ and $\delta_2$ of $(64,32)$ polar code constructed by PW with $d_{\min} = 8$ and $A_{d_{\min}} = 664$.
• Figure 2: Fig. \ref{['FigExampleMWD']}(a), Fig. \ref{['FigExampleMWD']}(b) and Fig. \ref{['FigExampleMWD']}(c) illustrate the examples of the nested MWD sequences with $N = 4, 8, 16$, respectively.
• Figure 3: The information set obtained by ECBS algorithm with $N=32$, $K=16$, $L=2$ and $E_b/N_0=1.25$dB.
• Figure 4: The BLER performance of polar codes with $N = 256$ and $K = 128$, where IMWD sequence means the inverse MWD sequence with the opposite criterion 3).
• Figure 5: Fig. \ref{['FigTargetSNR']}(a) and Fig. \ref{['FigTargetSNR']}(b) illustrate the minimum required SNRs of polar codes decoded by SCL decoding with $L = 8$ and $L = 16$ to achieve BLER $\le 10^{-4}$ and BLER $\le 10^{-3}$ under the AWGN channel with $N = 128$ and $N = 256$, respectively.

Theorems & Definitions (18)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1
• Definition 2
• Definition 3
• Lemma 3
• proof
• Lemma 4