Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
<|reference_start|>Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance: 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 \rho(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 \rho(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.<|reference_end|>
@article{yekhanin2004long,
    title={Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any
  Fixed Distance},
    author={Sergey Yekhanin, Ilya Dumer},
    journal={arXiv preprint arXiv:cs/0406039},
    year={2004},
    archivePrefix={arXiv},
    eprint={cs/0406039},
    primaryClass={cs.IT math.IT}
}