Complete Enumeration of Stopping Sets of Full-Rank Parity-Check Matrices of Hamming Codes

TL;DR

This paper derives a general formula for counting stopping sets of any size in full-rank parity-check matrices of Hamming codes, which is crucial for understanding their decoding performance over erasure channels.

Contribution

It extends previous work by providing a comprehensive enumeration of stopping sets of all sizes for Hamming code parity-check matrices.

Findings

Derived a formula for the number of stopping sets of any size

Generalizes previous results for size three stopping sets

Enhances understanding of Hamming code decoding performance

Abstract

Stopping sets, and in particular their numbers and sizes, play an important role in determining the performance of iterative decoders of linear codes over binary erasure channels. In the 2004 Shannon Lecture, McEliece presented an expression for the number of stopping sets of size three for a full-rank parity-check matrix of the Hamming code. In this correspondence, we derive an expression for the number of stopping sets of any given size for the same parity-check matrix.