Bounds on the minimum code distance for nonbinary codes based on bipartite graphs

Abstract

The minimum distance of codes on bipartite graphs (BG codes) over GF(q) is studied. A new upper bound on the minimum distance of BG codes is derived. The bound is shown to lie below the Gilbert-Varshamov bound when q ≤ 32. Since the codes based on bipartite expander graphs (BEG codes) are a special case of BG codes and the resulting bound is valid for any BG code, it is also valid for BEG codes. Thus, nonbinary (q ≤ 32) BG codes are worse than the best known linear codes. This is the key result of the work. We also obtain a lower bound on the minimum distance of BG codes with a Reed-Solomon constituent code and a lower bound on the minimum distance of low-density parity-check (LDPC) codes with a Reed-Solomon constituent code. The bound for LDPC codes is very close to the Gilbert-Varshamov bound and lies above the upper bound for BG codes.

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Correspondence to A. A. Frolov.

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Original Russian Text © A.A. Frolov, V.V. Zyablov, 2011, published in Problemy Peredachi Informatsii, 2011, Vol. 47, No. 4, pp. 27–42.

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Frolov, A.A., Zyablov, V.V. Bounds on the minimum code distance for nonbinary codes based on bipartite graphs. Probl Inf Transm 47, 327–341 (2011). https://doi.org/10.1134/S0032946011040028

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Keywords

  • Bipartite Graph
  • Information Transmission
  • Linear Code
  • LDPC Code
  • Maximum Distance Separable