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 \(\{1,2,\ldots,n\}\) 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 \(m \geq 3\).