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Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance

Sergey Yekhanin·Ilya Dumer
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Abstract

Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance d. We study the minimum asymptotic redundancy ρ(q,n,d)=n-log_q A(q,n,d) as n grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy ρ(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances 4,5 and 6.

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Sergey Yekhanin, Ilya Dumer. 2004-06-23. Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance. https://arxiv.org/abs/cs/0406039

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