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

TL;DR

This paper introduces long algebraic codes that asymptotically outperform BCH codes and the Gilbert-Varshamov bound for any fixed distance, reducing redundancy as code length increases.

Contribution

It presents a new construction of algebraic codes that surpass existing bounds for all fixed distances, extending previous results limited to small distances.

Findings

Codes improve on BCH codes for fixed distances

Asymptotic redundancy is minimized to known lower bounds

Surpass Gilbert-Varshamov bound for all fixed distances

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 \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.