An Improvement of the Gilbert-Varshamov Bound Over Nonprime Fields

The Gilbert–Varshamov bound guarantees the existence of families of codes over the finite field F` with good asymptotic parameters. We show that this bound can be improved for all non-prime fields F` with ` ≥ 49, except possibly ` = 125. We observe that the same improvement even holds within the class of transitive codes and within the class of self-orthogonal codes. The Gilbert–Varshamov bound guarantees the existence of families of codes over the finite field F` with good asymptotic parameters (information rate and relative minimum distance). In case ` ≥ 49 is a square, the bound was improved by the famous Tsfasman–Vlăduţ–Zink bound [12], using Goppa’s algebraic geometry codes and modular curves with many rational points over F`. Also, for ` = p with odd n > 1 and very large p (depending on n), there are improvements of the GV bound due to Niederreiter and Xing [9]. For a linear code C we denote by n(C), k(C) and d(C) its length, dimension and minimum distance. By R(C) = k(C)/n(C) and δ(C) = d(C)/n(C) we denote the information rate and the relative minimum distance of C, respectively. Following Manin [8], we define the set U` ⊆ R to be the set of all points (δ,R) such that there exists a family of codes (Ci)i≥0 over F` with n(Ci) → ∞, δ(Ci) → δ and R(Ci) → R, as i→∞. Manin proved that there exists a function α` : [0, 1]→ [0, 1] such that U` = {(δ,R) ∈ R | 0 ≤ δ ≤ 1, 0 ≤ R ≤ α`(δ) }. This function α`(δ) is continuous and non-increasing, and one knows that α`(0) = 1 and α`(δ) = 0 for 1− `−1 ≤ δ ≤ 1. All other values of α`(δ) are unknown. The explicit description of the function α`(δ) is considered to be one of the most important (and most difficult) problems in coding theory. Many upper bounds for α`(δ) are known, among them the (asymptotic) Plotkin bound and the linear programming bound, see [6] and [7]. One may argue that lower bounds are more important since every non-trivial lower bound for α`(δ) assures the existence of long codes over F` having good parameters. The classical lower bound for α`(δ) is the Gilbert–Varshamov bound (GV bound) which states that α`(δ) ≥ 1− δ log`(`− 1) + δ log`(δ) + (1− δ) log`(1− δ), for all δ ∈ (0, 1− `−1). (1) ∗Alp Bassa is supported by Tübitak Proj. No. 112T233 †Peter Beelen is supported by DNRF (Denmark) and NSFC (China), grant No.11061130539. ‡Arnaldo Garcia is supported by CNPq (Brazil) and Sabancı University (Turkey). §Henning Stichtenoth is supported by Tübitak Proj. No. 111T234.