An Improvement of the Gilbert–Varshamov Bound Over Nonprime Fields

Abstract

The Gilbert-Varshamov bound guarantees the existence of families of codes over the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sub> with good asymptotic parameters. We show that this bound can be improved for all nonprime fields F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sub> with ℓ ≥ 49 , except possibly ℓ = 125. We observe that the same improvement even holds within the class of transitive codes and within the class of self-orthogonal codes.