New quasi-twisted codes over F₁₁— minimum distance bounds and a new database

Abstract

One fundamental and challenging problem in coding theory is to optimize the parameters [n, k, d] of a linear code over the finite field Fq and construct codes with best possible parameters. There are tables and databases of best-known linear codes over the finite fields of size up to 9 together with upper bounds on the minimum distances. Motivated by recent works on codes over F 11 , we present a table of best-known linear codes over F 11 together with upper bounds on minimum distances. Our table covers the range n ≤ 150 for the length, and 3 ≤ k ≤ 7 for the dimension. To the best of our knowledge, this is the first time such a table is presented in the literature. For the construction of the best-known codes, we employed an iterative heuristic search algorithm to search for new linear codes in the class of quasi-twisted (QT) codes. The search yielded many new codes with better parameters than previously known codes. In many cases, optimal codes are obtained. In addition to presenting a comprehensive table of best-known codes over F 11 of dimensions up to 7 with upper bounds on the minimum distances, we also present separate tables for the optimal codes and new QT codes over F 11 . We hope that this work will be a useful source for further study on codes over F 11 .

Construction of quasi-twisted codes and enumeration of defining polynomials

“…A code which achieves either of these values is called optimal. Tables of best known linear codes exist for q = 2 to 9 [6], 11 [3] and 13 [4].…”

Generating generalized necklaces and new quasi-cyclic codes

“…Venkaiah and Gulliver [31] constructed quasi-cyclic [pk, k, d] 13 codes of dimensions k ≤ 6 and n ≤ 150. In [12] and [13] E. Chen and N. Aydin constructed 45 new OC and QT codes over GF (11), 38 such codes over GF (13) and presented databases for small dimensions and n ≤ 150. New QT codes over GF (11) and GF (13) are also presented in [7].…”

New Linear Codes over GF(3), GF(11), and GF(13)

“…Computer algebra system Magma also contains a similar database [8]. More recently, tables of best known codes over GF (11) and GF (13) have been published in [10] and [9].…”