Stopping sets of binary parity-check matrices with constant weight columns and stopping redundancy of the associated codes

Abstract

The binary linear code H_m,q, m> q≥2, of length ( _q^m) presented by a parity-check matrix H_m ^q whose columns are all distinct strings of length m and Hamming weight q ≥ 2 is considered. It is shown that the stopping distance of these matrices is three. A closed-form formula for the number of stopping sets of arbitrary size associated with these parity-check matrices is given. It turns out that H_m,_2q with m ≥ 3q is an optimal redundancy code, that is redundancy and stopping redundancy of this code are equal. Among the H_m,3 codes the only optimal redundancy code is H_5,3. It is shown that the stopping redundancy of H^m,2q+1 is bounded above by 2m - 2.