On codes from hypergraphs

• Published in: European journal of combinatorics Vol. 25; no. 3; pp. 339 - 354
• Main Authors: Bilu, Yonatan, Hoory, Shlomo
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2004
• ISSN: 0195-6698; 1095-9971
• DOI: 10.1016/j.ejc.2003.10.002

We propose a new family of asymptotically good binary codes, generalizing previous constructions of expander codes to t-uniform hypergraphs. We also describe an efficient decoding algorithm for these codes, that for a certain region of rates improves the known results for decoding distance of expander codes. The construction is based on hypergraphs with a certain “expansion” property called herein ϵ-homogeneity. For t-uniform t-partite Δ-regular hypergraphs, the expansion property required is roughly as follows: given t sets, A 1,…, A t , one at each side, the number of hyper-edges with one vertex in each set is approximately what would be expected had the edges been chosen at random. We show that in an appropriate random model, almost all hypergraphs have this property, and also present an explicit construction of such hypergraphs. Having a family of such hypergraphs, and a small code C 0⊆{0,1} Δ , with relative distance δ 0 and rate R 0, we construct “hypergraphs codes”. These have rate ≥ tR 0−( t−1), and relative distance ≥ δ 0 t/( t−1) − o(1). When t=2 l we also suggest a decoding algorithm, and prove that the fraction of errors that it decodes correctly is at least 2l−1 l −1/l·(δ 0/2) (l+1)/l−o(1) . In both cases, the o(1) is an additive term that tends to 0 as the length of the hypergraph code tends to infinity.