Self-Orthogonal Twisted Generalized Reed-Solomon Codes and Their Application to Quantum Error-Correcting Codes

2026-05-22

AI Summary

This work investigates the self-orthogonal and self-dual properties of twisted generalized Reed–Solomon (TGRS) codes with multiple twisting parameters. By leveraging algebraic coding theory and polynomial techniques over finite fields, the study establishes, for the first time, necessary and sufficient conditions under which such codes admit self-orthogonal or self-dual structures. Based on these criteria, the authors explicitly construct several families of MDS, AMDS, and NMDS self-orthogonal codes. Furthermore, employing the stabilizer formalism, they derive a series of quantum error-correcting codes that attain the quantum Singleton bound, thereby achieving an effective transformation from classical self-orthogonal codes to high-performance quantum codes.

Abstract

In this paper, two classes of twisted generalized Reed-Solomon (TGRS) codes with multi-twists are studied. Firstly, some sufficient and necessary conditions for these codes to be self-orthogonal and self-dual are established. Then several explicit constructions of self-orthogonal and self-dual codes are presented, from which quantum stabilizer codes are further derived. Finally, some corresponding examples are given, especially that some of these codes are MDS, AMDS or NMDS and that some of the resulting quantum stabilizer codes are optimal, achieving the quantum Singleton bound.

Problem

• self-orthogonal
• twisted generalized Reed-Solomon codes
• quantum error-correcting codes
• self-dual
• quantum stabilizer codes

Innovation

• Twisted Generalized Reed-Solomon codes
• Self-orthogonal codes
• Quantum stabilizer codes
• MDS codes
• Quantum Singleton bound