On the Minimum Distance of Low-Density Parity-Check Codes with Parity-Check Matrices Constructed from Permutation Matrices

An ensemble of codes defined by parity-check matrices composed of $M\times M$ 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 $M\to\infty$, 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].