Enumeration of Minimum Hamming Weight Polar Codewords with Sublinear Complexity

TL;DR

This paper introduces a sublinear-complexity method for enumerating minimum Hamming weight polar codewords, leveraging their generation by successive cancellation list decoding and features of zero-capacity bit-channels.

Contribution

It proposes a novel, efficient strategy for enumerating and counting minimum Hamming weight polar codewords with sublinear complexity, improving over existing methods.

Findings

The upper bound for the number of PC-MHW is tight and often exact.

The proposed method requires less than half the list size of previous algorithms.

The approach is validated through theoretical analysis and practical enumeration results.

Abstract

Polar code, with explicit construction and recursive structure, is the latest breakthrough in channel coding field for its low-complexity and theoretically capacity-achieving property. Since polar codes can approach the maximum likelihood performance under successive cancellation list decoding (SCLD), its decoding performance can be evaluated by Bonferroni-type bounds (e.g., union bound) in which the Hamming weight spectrum will be used. Especially, the polar codewords with minimum Hamming weight (PC-MHW) are the most important item in that bound because they make major contributions to the decoding error pattern particularly at high signal-to-noise-ratio. In this work, we propose an efficient strategy for enumerating the PC-MHW and its number. By reviewing the inherent reason that PC-MHW can be generated by SCLD, we obtain some common features of PC-MHW captured by SCLD. Using these features, we introduce a concept of zero-capacity bit-channel to obtain a tight upper bound for the number of PC-MHW, whose computing complexity is sublinear with code length. Furthermore, we prove that the proposed upper bound is really the exact number of PC-MHW in most cases. Guided by the bound and its theoretical analysis, we devise an efficient SCLD-based method to enumerate PC-MHW, which requires less than half of the list size compared with the existing methods.