Stopping Sets of Binary Parity-check Matriceswith Constant Weight Columns and StoppingRedundancy of the Associated Codes

摘要

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 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.