New Codes from Cyclic and Negacyclic Codes of Even Length over $\mathbb{Z}_4$

2026-06-07

AI Summary

This study addresses the construction of linear codes with improved parameters and high-performance binary quantum codes by systematically investigating cyclic and negacyclic codes over even-length quaternary rings for the first time. Leveraging algebraic coding theory and structural analysis, the authors develop an efficient computer-aided search algorithm that identifies 2,500 new cyclic codes and 730 new negacyclic codes, all of which surpass the best-known codes in terms of parameters. These newly discovered codes further enable the derivation of a collection of binary quantum codes with excellent parameters, significantly expanding the current repertoire of available codes for quantum error correction.

Abstract

This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over $\mathbb{Z}_4$ from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such $\mathbb{Z}_4$ codes.

Problem

• cyclic codes
• negacyclic codes
• linear codes over Z4
• binary quantum codes
• code parameters

Innovation

• cyclic codes
• negacyclic codes
• ℤ₄-linear codes
• quantum codes
• code search algorithm