Binary optimal linear codes with various hull dimensions and entanglement-assisted QECC

TL;DR

This paper introduces a new construction method for binary linear codes with various hull dimensions, enabling the creation of optimal codes up to length 13 and applying them to develop high-performance entanglement-assisted quantum error-correcting codes.

Contribution

A novel building-up construction method for binary codes with specified hull dimensions, leading to the classification of optimal codes and their application in quantum error correction.

Findings

Constructed all optimal binary codes with hull dimensions 1 and 2 up to length 13.

Constructed all optimal binary codes with hull dimensions 3, 4, 5 up to length 12.

Developed entanglement-assisted quantum codes with the best known parameters.

Abstract

The hull of a linear code $C$ is the intersection of $C$ with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension $\ge 2$ except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary $[n+2, k+1]$ codes with hull dimension $\ell, \ell +1$, or $\ell +2$ from a given binary $[n,k]$ code with hull dimension $\ell$. In particular, with respect to hull dimensions 1 and 2, we construct all binary optimal $[n, k]$ codes of lengths up to 13. With respect to hull dimensions 3, 4, and 5, we construct all binary optimal $[n,k]$ codes of lengths up to 12 and the best possible minimum distances of $[13,k]$ codes for $3 \le k \le 10$. As an application, we apply our binary optimal codes with a given hull dimension to construct several entanglement-assisted quantum error-correcting codes(EAQECC) with the best known parameters.