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A Hybrid Reliability--Weight Framework for Construction of Polar Codes

Mohammad Rowshan, Vlad-Florin Dragoi

TL;DR

This work addresses finite-length polar-code design by blending reliability-based bit-channel ordering with algebraic distance information from decreasing monomial codes. It introduces a K-dependent mixed per-bit cost that combines SC reliability with a distance penalty derived from orbit-based minimum-weight contributions, and shows this design minimizes a truncated SC/ML union bound while behaving as a local perturbation to purely reliability-based sets. Sub-maximal degree monomials are shown to be asymptotically negligible, and the mixed metric aligns with near-ML decoding via a bit-wise error-event decomposition for SC and SCL decoding. Numerical results on BPSK-AWGN demonstrate significant finite-length gains in the minimum distance and its multiplicity with modest reliability loss, clarifying the practical impact for near-ML decoders and short-to-moderate blocklengths.

Abstract

Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.

A Hybrid Reliability--Weight Framework for Construction of Polar Codes

TL;DR

This work addresses finite-length polar-code design by blending reliability-based bit-channel ordering with algebraic distance information from decreasing monomial codes. It introduces a K-dependent mixed per-bit cost that combines SC reliability with a distance penalty derived from orbit-based minimum-weight contributions, and shows this design minimizes a truncated SC/ML union bound while behaving as a local perturbation to purely reliability-based sets. Sub-maximal degree monomials are shown to be asymptotically negligible, and the mixed metric aligns with near-ML decoding via a bit-wise error-event decomposition for SC and SCL decoding. Numerical results on BPSK-AWGN demonstrate significant finite-length gains in the minimum distance and its multiplicity with modest reliability loss, clarifying the practical impact for near-ML decoders and short-to-moderate blocklengths.

Abstract

Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.
Paper Structure (25 sections, 4 theorems, 39 equations, 1 table)

This paper contains 25 sections, 4 theorems, 39 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Structural Tools
  3. Notation and union bound
  4. Polar codes and bit-channel reliabilities
  5. Monomial representation and decreasing codes
  6. Minimum distance and orbit-based $A_{w_{\min}}$
  7. Mixed Reliability--Weight Design
  8. Bit-wise ML contribution and local distance
  9. Monomial-level weight penalty at fixed $K$
  10. Channel-dependent distance penalty and mixed cost
  11. K-dependent mixed construction
  12. ML-negligibility of sub-maximal degrees
  13. Bit-wise Mixed Metric for Near-ML Decoders
  14. SC first-error events
  15. Per-bit split of a truncated ML union bound
  16. ...and 10 more sections

Key Result

Lemma 1

Let $W$ be a fixed BMS channel. Assume there exist constants $c(W)>0$ and $\gamma(W)\in(0,1)$ such that $P_w(W) \le c(W)\gamma(W)^w$ for all $w\ge1$. Let $N=2^m$ and $w_{\min}=2^{m-r}$ be the minimum distance of a decreasing monomial code with maximum degree $r$. If for some $j$, then

Theorems & Definitions (11)

  • Definition 1: Monomial and decreasing monomial codes
  • Definition 2: Bit-wise minimum distance and ML contribution
  • Definition 3: Structural weight penalty
  • Definition 4: Mixed monomial cost
  • Definition 5: K-dependent mixed design
  • Lemma 1: Sub-maximal degrees are ML-negligible
  • Definition 6: Pruning and ML-like events for SCL
  • Lemma 2: SCL block-error decomposition
  • Definition 7: Local perturbation
  • Lemma 3: Effect on SC sum
  • ...and 1 more