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Bounds on the minimum code distance for nonbinary codes based on bipartite graphs. (English. Russian original) Zbl 1302.94071
Probl. Inf. Transm. 47, No. 4, 327-341 (2011); translation from Probl. Peredachi Inf. 47, No. 4, 27-42 (2011).
Summary: The minimum distance of codes on bipartite graphs (BG codes) over $$\mathrm{GF}(q)$$ 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 $$q\geq 32$$. 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 $$(q\geq 32)$$ 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.

##### MSC:
 94B60 Other types of codes 94B65 Bounds on codes
Full Text:
##### References:
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