Optimal quaternary linear codes with one-dimensional Hermitian hull and the related EAQECCs

TL;DR

This paper determines the maximum minimum distance of certain quaternary linear codes with one-dimensional Hermitian hulls for small lengths and dimensions, solves a related conjecture, and constructs improved quantum error-correcting codes.

Contribution

It develops a method to exactly compute the maximum minimum distance for specific quaternary codes with one-dimensional Hermitian hulls and constructs new optimal or improved EAQECCs.

Findings

Exact values of $D_4^H(n,k,1)$ for specified parameters are determined.

A conjecture on the maximum minimum distance is proven.

New binary EAQECCs with optimal or improved parameters are constructed.

Abstract

Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact value of $D_4^H(n,k,1)$ for $n\leq 12$ or $k\in \{1,2,3,n-1,n-2,n-3\}$, where $D_4^H(n,k,1)$ denotes the largest minimum distance among all quaternary linear $[n,k]$ codes with one-dimensional Hermitian hull. As a consequence, we solve a conjecture proposed by Mankean and Jitman on the largest minimum distance of a quaternary linear code with one-dimensional Hermitian hull. As an application, we construct some binary entanglement-assisted quantum error-correcting codes (EAQECCs) from quaternary linear codes with one-dimensional Hermitian hull. Some of these EAQECCs are optimal codes, and some of them are better than previously known ones.