Some best rate 1/p and rate (p-1)/p systematic quasi-cyclic codes over GF(3) and GF(4)

Abstract

The class of quasi-cyclic (QC) codes has been proven to contain many good codes. To date the known QC codes are primarily rate 1/p and (p-1)/p binary codes constructed from circulant matrices. These results are extended to QC codes over GF(3) and GF(4). Codes are constructed using integer linear programming and heuristic combinatorial optimization. Many of these attain the maximum possible minimum distance for any linear code with the same parameters, and several improve the maximum known distances. The link between power residue (PR) codes and QC codes is exploited as a means of constructing new QC codes and to initialize the search algorithm. Previously unknown minimum distances for nonbinary quadratic residue codes are given. The minimum distances for the PR codes and the maximum known distances for the QC codes are tabulated.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>