Generalized Spectral Bound for Quasi-Twisted Codes
Buket Özkaya
TL;DR
This paper extends the generalized spectral bound framework to quasi-twisted codes, enabling the use of multiple defining-set bounds to obtain tighter lower bounds on the minimum distance than existing approaches such as the Jensen bound. By reformulating QT codes within the spectral paradigm—through eigenvalues, eigenspaces, and eigencodes—and by leveraging a concatenated-code perspective, the authors derive a generalized spectral bound that subsumes prior spectral results. The approach yields improved distance estimates in several examples and is supported by MAGMA-based simulations, illustrating practical gains in QT-code performance. The work offers a unified spectral toolkit for QT codes that can outperform traditional bounds and broadens the applicability of spectral methods in algebraic coding theory.
Abstract
Semenov and Trifonov [22] developed a spectral theory for quasi-cyclic codes and formulated a BCH-like minimum distance bound. Their approach was generalized by Zeh and Ling [24], by using the HT bound. The first spectral bound for quasi-twisted codes appeared in [7], which generalizes Semenov-Trifonov and Zeh-Ling bounds, but its overall performance was observed to be worse than the Jensen bound. More recently, an improved spectral bound for quasi-cyclic codes was proposed in [15], which outperforms the Jensen bound in many cases. In this paper, we adopt this approach to quasi-twisted case and we show that this new generalized spectral bound provides tighter lower bounds on the minimum distance compared to the Jensen and Ezerman et. al. bounds.