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

TL;DR

This paper introduces a hybrid reliability-weight framework for constructing polar codes that balances capacity-achieving reliability with improved minimum-weight spectra at short and moderate lengths.

Contribution

It combines algebraic and reliability insights into a mixed bit-channel ordering, optimizing code performance by minimizing a union-bound surrogate.

Findings

Hybrid construction improves minimum distance and weight spectra.

Numerical results show trade-offs between reliability and weight spectrum.

Asymptotic impact of mixed design vanishes for large code lengths.

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.