Computational Complexity
Introduction to Coding Theory
Expander-Based Constructions of Efficiently Decodable Codes
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Computationally efficient error-correcting codes and holographic proofs
Computationally efficient error-correcting codes and holographic proofs
IEEE Transactions on Information Theory - Part 1
IEEE Transactions on Information Theory
Error exponents of expander codes
IEEE Transactions on Information Theory
A Randomness-Efficient Sampler for Matrix-valued Functions and Applications
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Low-complexity error correction of Hamming-code-based LDPC codes
Problems of Information Transmission
On the number of errors correctable with codes on graphs
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
Woven graph codes: asymptotic performances and examples
IEEE Transactions on Information Theory
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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, A1,...,At, 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 C0 ⊆ {0, 1}Δ, with relative distance δ0 and rate R0, we construct "hypergraphs codes". These have rate ≥ tR0 - (t - 1), and relative distance ≥ δ0t/(t-1) - o(1). When t = 2l 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 ċ (δ0/2)(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.