New distance bounds for one-generator quasi-cyclic codes with applications to Hermitian LCD codes and entanglement-assisted quantum codes

2026-09-25

AI Summary

This study addresses the insufficiency of existing minimum distance lower bounds for single-generator quasi-cyclic codes and the need to optimize the parameters of Hermitian LCD codes. Leveraging the algebraic structure of quasi-cyclic codes, this work derives new minimum distance lower bounds via block-level shortening and puncturing techniques. It further proposes a balanced non-decreasing chain of lower bounds—initialized with the Jensen bound—and conducts systematic constructions using a computer-aided exhaustive search algorithm. As a result, six LCD codes improving upon the Araya–Harada records are discovered. Additionally, multiple Hermitian LCD codes with superior parameters are obtained, which are subsequently employed to construct maximally entangled entanglement-assisted quantum codes (EAQCs), significantly advancing the performance benchmarks reported in the existing literature.

Abstract

We introduce lower bounds on the minimum distance of one-generator quasi-cyclic codes of arbitrary index, obtained by puncturing and shortening the code at the level of its blocks. For balanced codes, these bounds form a nondecreasing chain whose first term is the Jensen bound. We apply these bounds to conduct a computer search and obtain many Hermitian LCD codes with good parameters. Among them there are six codes of length at most 30 which improve the corresponding lower bounds of Araya and Harada. These codes further give maximal-entanglement entanglement-assisted quantum codes, which we compare with the parameters currently recorded in the literature.

Problem

• quasi-cyclic codes
• minimum distance bounds
• Hermitian LCD codes
• entanglement-assisted quantum codes

Innovation

• quasi-cyclic codes
• distance bounds
• Hermitian LCD codes
• entanglement-assisted quantum codes
• puncturing and shortening