Hermitian Self-dual Twisted Generalized Reed-Solomon Codes

TL;DR

This paper characterizes Hermitian self-dual twisted generalized Reed-Solomon (TGRS) codes, providing new constructions and conditions for their existence, which enhances understanding of their structure and potential applications in coding theory.

Contribution

It introduces a unified framework for Hermitian self-dual TGRS codes, offering new constructions, conditions, and classes of such codes with flexible parameters.

Findings

Established necessary and sufficient conditions for Hermitian self-duality.

Presented four new constructions of self-dual TGRS codes.

Derived conditions for TGRS codes to be both Hermitian self-dual and MDS.

Abstract

Self-dual maximum distance separable (MDS) codes over finite fields are linear codes with significant combinatorial and cryptographic applications. Twisted generalized Reed-Solomon (TGRS) codes can be both MDS and self-dual. In this paper, we study a general class of TGRS codes (A-TGRS), which encompasses all previously known special cases. First, we establish a sufficient and necessary condition for an A-TGRS code to be Hermitian self-dual. Furthermore, we present four constructions of self-dual TGRS codes, which, to the best of our knowledge, nearly cover all the related results previously reported in the literature. More importantly, we also obtain several new classes of Hermitian self-dual TGRS codes with flexible parameters. Based on this framework, we derive a sufficient and necessary condition for an A-TGRS code to be Hermitian self-dual and MDS. In addition, we construct a class of MDS Hermitian self-dual TGRS code by appropriately selecting the evaluation points. This work investigates the Hermitian self-duality of TGRS codes from the perspective of matrix representation, leading to more concise and transparent analysis. More generally, the Euclidean self-dual TGRS codes and the Hermitian self-dual GRS codes can also be understood easily from this point.