Decoding of Expander Codes at Rates Close to Capacity

The decoding error probability of codes is studied as a function of their block length. It is shown that the existence of codes with a polynomially small decoding error probability implies the existence of codes with an exponentially small decoding error probability. Specifically, it is assumed that there exists a family of codes of length N and rate R=(1-\epsilon)C (C is a capacity of a binary symmetric channel), whose decoding probability decreases polynomially in 1/N. It is shown that if the decoding probability decreases sufficiently fast, but still only polynomially fast in 1/N, then there exists another such family of codes whose decoding error probability decreases exponentially fast in N. Moreover, if the decoding time complexity of the assumed family of codes is polynomial in N and 1/\epsilon, then the decoding time complexity of the presented family is linear in N and polynomial in 1/\epsilon. These codes are compared to the recently presented codes of Barg and Zemor, ``Error Exponents of Expander Codes,'' IEEE Trans. Inform. Theory, 2002, and ``Concatenated Codes: Serial and Parallel,'' IEEE Trans. Inform. Theory, 2005. It is shown that the latter families can not be tuned to have exponentially decaying (in N) error probability, and at the same time to have decoding time complexity linear in N and polynomial in 1/\epsilon.Comment: Appears in IEEE Transactions on Information Theory, December 2006. The short version of this paper appears in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

Distance properties of expander codes

We study the minimum distance of codes defined on bipartite graphs. Weight spectrum and the minimum distance of a random ensemble of such codes are computed. It is shown that if the vertex codes have minimum distance $\ge 3$, the overall code is asymptotically good, and sometimes meets the Gilbert-Varshamov bound. Constructive families of expander codes are presented whose minimum distance asymptotically exceeds the product bound for all code rates between 0 and 1.Comment: 19 pages, 7 figure

On Nearly-MDS Expander Codes

A construction of expander codes is presented with the following three properties: (i) the codes lie close to the Singleton bound, (ii) they can be encoded in time complexity that is linear in their code length, and (iii) they have a linear-time bounded-distance decoder. By using a version of the decoder that corrects also erasures, the codes can replace MDS outer codes in concatenated constructions, thus resulting in linear-time encodable and decodable codes that approach the Zyablov bound or the capacity of memoryless channels. The presented construction improves on an earlier result by Guruswami and Indyk in that any rate and relative minimum distance that lies below the Singleton bound is attainable for a significantly smaller alphabet size

On Nearly-MDS Expander Codes 1

Abstract — A construction of graph codes is presented that approaches the Singleton bound as the alphabet size goes to infinity. These codes can be decoded by a combined error-erasure decoder whose time complexity grows linearly with the code length. I