Abstract
Error control codes have been widely used in data communications and storage systems. One central problem in coding theory is to optimize the parameters of a linear code and construct codes with best possible parameters. There are tables of best-known linear codes over finite fields of sizes up to 9. Recently, there has been a growing interest in codes over $\mathbb{F}_{13}$ and other fields of size greater than 9. The main purpose of this work is to present a database of best-known linear codes over the field $\mathbb{F}_{13}$ together with upper bounds on the minimum distances. To find good linear codes to establish lower bounds on minimum distances, an iterative heuristic computer search algorithm is employed to construct quasi-twisted (QT) codes over the field $\mathbb{F}_{13}$ with high minimum distances. A large number of new linear codes have been found, improving previously best-known results. Tables of $[pm, m]$ QT codes over $\mathbb{F}_{13}$ with best-known minimum distances as well as a table of lower and upper bounds on the minimum distances for linear codes of length up to 150 and dimension up to 6 are presented.
Highlights
Introduction and motivationLet [n, k, d]q denote a linear code of length n, dimension k and minimum distance d over the finite field Fq
The main purpose of this work is to present a database of best-known linear codes over the field F13 together with upper bounds on the minimum distances
We present a database of linear codes over F13 for lengths ≤ 150 and dimensions 3 ≤ k ≤ 6
Summary
Let [n, k, d]q denote a linear code of length n, dimension k and minimum distance (weight) d over the finite field Fq. The paper contains several tables: tables of new, improved QC and QT codes, maximum known minimum distances for QT [pm, m] codes, optimal QT codes, as well as a comprehensive table of lower and upper bounds on linear codes over F13 that covers the range n ≤ 150 and 3 ≤ k ≤ 6. With these concrete results, this work can serve as a foundation for future research on linear codes over F13 (e.g. a more comprehensive database)
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