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On the minimum distance of low-density parity-check codes with parity-check matrices constructed from permutation matrices. (English. Russian original) Zbl 1078.94038
Probl. Inf. Transm. 41, No. 1, 33-44 (2005); translation from Probl. Peredachi Inf. 41, No. 1, 39-52 (2005).
Summary: 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 R. G. Gallager [Low-density parity-check codes, Cambridge: MIT Press (1963)]. 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.

MSC:
94B60 Other types of codes
94A45 Prefix, length-variable, comma-free codes
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