A Comparison of Distance Bounds for Quasi-Twisted Codes

TL;DR

This paper introduces spectral bounds for the minimum distance of quasi-twisted codes, generalizing existing bounds and comparing their effectiveness through spectral theory and code structure analysis.

Contribution

It proposes new spectral bounds for quasi-twisted codes and compares their performance with existing bounds, revealing conditions where some bounds outperform others.

Findings

Jensen bound always outperforms spectral bound under certain conditions

Spectral bounds generalize Semenov-Trifonov and Zeh-Ling bounds

Performance comparison shows relative strengths of Lally, Jensen, and spectral bounds

Abstract

Spectral bounds on the minimum distance of quasi-twisted codes over finite fields are proposed, based on eigenvalues of polynomial matrices and the corresponding eigenspaces. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a way similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are presented in comparison with each other.