Efficiently decodable codes meeting Gilbert-Varshamov bound for low rates
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We demonstrate a probabilistic construction of binary linear codes meeting the GV bound (with overwhelming probability) for rates up to about 10-4 together with polynomial time algorithms to perform encoding and decoding up to half the distance. The only previous result of this type (for rates up to about 0.02) suffered from sub-exponential time decoding [3].
[1] Christian Thommesen. The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound , 1983, IEEE Trans. Inf. Theory.
[2] Venkatesan Guruswami,et al. Decoding concatenated codes using soft information , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.