Matrix product and quasi-twisted codes in one class

TL;DR

This paper extends matrix product codes to a broader class called generalized matrix product (GMP) codes, unifies QT codes within this framework, and introduces new bounds on their minimum distance with practical examples.

Contribution

It introduces GMP codes that encompass QT codes, provides generator matrix formulas, and develops new bounds on minimum distance, expanding the theoretical understanding of these code classes.

Findings

GMP codes include QT codes as a special case.

New lower bounds on the minimum distance of GMP codes.

Examples show bounds are tight and some GMP codes outperform known codes.

Abstract

Many classical constructions, such as Plotkin's and Turyn's, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Despite that the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.