On the Stopping Distance and the Stopping Redundancy of Codes

TL;DR

This paper explores the concept of stopping redundancy in linear codes, establishing bounds and constructions to optimize the stopping distance, which impacts iterative decoding performance.

Contribution

It introduces the stopping redundancy parameter, providing bounds and analyzing its behavior for various code families, including Reed-Muller and Golay codes.

Findings

Stopping redundancy can be minimized to match the code's minimum distance.

For Reed-Muller codes, stopping redundancy is at most a constant times their traditional redundancy.

Golay codes have bounded stopping redundancies of at most 35 (binary) and 22 (ternary).

Abstract

It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$. Several recent papers refer to this parameter as the \emph{stopping distance} $s$ of $C$. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for $C$ depends on the corresponding choice of a parity-check matrix. It is easy to see that $s \le d$, where $d$ is the minimum Hamming distance of $C$, and we show that it is always possible to choose a parity-check matrix for $C$ (with sufficiently many dependent rows) such that $s = d$. We thus introduce a new parameter, termed the \emph{stopping redundancy} of $C$, defined as the minimum number of rows in a parity-check matrix $H$ for $C$ such that the corresponding stopping distance $s(H)$ attains its largest possible value, namely $s(H) = d$. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.