Refined Upper Bounds on Stopping Redundancy of Binary Linear Codes

TL;DR

This paper refines upper bounds on the stopping redundancy of binary linear codes by improving probabilistic analysis and strategic row selection in parity-check matrices, leading to tighter bounds than previous methods.

Contribution

It introduces an improved probabilistic approach and row selection strategy to better estimate stopping redundancy bounds for binary linear codes.

Findings

Enhanced bounds on stopping redundancy $ ho_l( ext{C})$ for all $l$

Comparison shows improved bounds over existing methods

Numerical results validate the effectiveness of the new approach

Abstract

The $l$-th stopping redundancy $\rho_l(\mathcal C)$ of the binary $[n, k, d]$ code $\mathcal C$, $1 \le l \le d$, is defined as the minimum number of rows in the parity-check matrix of $\mathcal C$, such that the smallest stopping set is of size at least $l$. The stopping redundancy $\rho(\mathcal C)$ is defined as $\rho_d(\mathcal C)$. In this work, we improve on the probabilistic analysis of stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best bounds known today. In our approach, we judiciously select the first few rows in the parity-check matrix, and then continue with the probabilistic method. By using similar techniques, we improve also on the best known bounds on $\rho_l(\mathcal C)$, for $1 \le l \le d$. Our approach is compared to the existing methods by numerical computations.