On the minimum distance of codes with parity-check matrices constructed from permutation matrices

An ensemble of codes defined by parity-check matrices composed of M × M permutation matrices is considered. This ensemble is a subensemble of the ensemble of low-density parity-check (LDPC) codes considered by Gallager [1]. We prove that, as M → ∞, the minimum distance of almost all codes in the ensemble grows linearly with M. We also show that in several cases the asymptotic minimum-distance-to-block-length ratio for almost all codes in the ensemble satisfies Gallager’s bound [1].