On Hull-Variation Problem of Equivalent Linear Codes

TL;DR

This paper studies the hull-variation problem of linear codes under equivalence transformations, introduces the maximal hull dimension as an invariant, and constructs new families of codes including LCD, negacyclic, BCH, and quantum error-correcting codes.

Contribution

It introduces the maximal hull dimension as a new invariant and applies it to classify and construct various classes of codes, including MDS and EAQEC codes.

Findings

Linear codes with any hull dimension are characterized via equivalence.

New families of LCD negacyclic and BCH codes over ${\bf F}_3$ are constructed.

Several new entanglement-assisted quantum codes, including MDS and almost MDS, are developed.

Abstract

The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer $h$ satisfying $0 \leq h \leq n-1$, a linear $[2n, n]_q$ self-dual code is equivalent to a linear $h$-dimension hull code. On the opposite direction we prove that a linear LCD code over ${\bf F}_{2^s}$ satisfying $d\geq 2$ and $d^{\perp} \geq 2$ is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over ${\bf F}_3$ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.