A Generalization of Quasi-twisted Codes: Multi-twisted codes

TL;DR

This paper introduces multi-twisted (MT) codes, a new generalization of quasi-twisted codes, demonstrating their potential to produce codes with superior parameters and exploring related algebraic properties and bounds.

Contribution

The paper defines multi-twisted codes, develops construction methods, and shows they can outperform existing QT and constacyclic codes, opening new avenues for code optimization.

Findings

Existence of codes with better parameters than known QT codes

Development of construction methods for MT codes

Discovery of a novel result on binomials over finite fields

Abstract

Cyclic codes and their various generalizations, such as quasi-twisted (QT) codes, have a special place in algebraic coding theory. Among other things, many of the best-known or optimal codes have been obtained from these classes. In this work we introduce a new generalization of QT codes that we call multi-twisted (MT) codes and study some of their basic properties. Presenting several methods of constructing codes in this class and obtaining bounds on the minimum distances, we show that there exist codes with good parameters in this class that cannot be obtained as QT or constacyclic codes. This suggests that considering this larger class in computer searches is promising for constructing codes with better parameters than currently best-known linear codes. Working with this new class of codes motivated us to consider a problem about binomials over finite fields and to discover a result that is interesting in its own right.