The Stopping Distance of Binary $BCH$-code Parity-check Matrices

Abstract

Given a parity-check matrix $H$ with $n$ columns, an $\ell$-subset $T$ of $\{1,2,\dots ,n\}$ is called a stopping set of size $\ell$ for $H$ if the $\ell$-column submatrix of $H$ consisting of columns with coordinate indexes in $T$ has no row of Hamming weight one. The size of the smallest non-empty stopping sets for $H$ is called the stopping distance of $H$.

In this paper, the stopping distance of $H_m(2t+1)$, parity-check matrices representing binary $t$-error-correcting $BCH$ codes, is addressed. It is shown that if $m$ is even then the stopping distance of this matrix is three. We conjecture that this property holds for all integers $m\ge 3$.