Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. There has been renewed interest in LCD codes due to an important application in cryptography. On the other hand, it is an important task to construct maximum-distance-separable (MDS) codes with good parameters in both coding theory and practice. The main purpose of this paper is to construct several classes of MDS codes with complimentary duals (i.e., LCD MDS codes). Then we give the new results on EAQEC MDS codes with maximal entanglement.Linear complementary-dual (LCD for short) codes are linear codes that intersect with their duals trivially. There has been renewed interest in LCD codes due to an important application in cryptography. On the other hand, it is an important task to construct maximum-distance-separable (MDS) codes with good parameters in both coding theory and practice. The main purpose of this paper is to construct several classes of MDS codes with complimentary duals (i.e., LCD MDS codes). Then we give the new results on EAQEC MDS codes with maximal entanglement.

Discrete Mathematics Algorithms and ApplicationsMay 15, 20260 citations

Constructions of quantum MDS codes

Hualu Liu, Xiusheng Liu

arXiv:2002.06040Quantum Information ProcessingJan 1, 202115 citations

Some constructions of quantum MDS codes

Simeon Ball

arXiv:1907.04391Designs Codes and CryptographyFeb 23, 202144 citations

Two new classes of Hermitian self-orthogonal non-GRS MDS codes and their applications

Gaojun Luo, Xiwang Cao, Martianus Frederic Ezerman +1 more

Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.Hermitian self-orthogonal codes form an essential ingredient in the construction of quantum stabilizer codes. In this paper, we present two new classes of Hermitian self-orthogonal MDS codes from twisted generalized Reed-Solomon codes. The constructed codes are monomially inequivalent to generalized Reed-Solomon codes. Two new classes of Hermitian LCD MDS codes can then be derived. Finally, based on the constructed Hermitian self-orthogonal MDS codes, we present two classes of quantum MDS codes with new parameters.

Advances in Mathematics of CommunicationsJan 1, 20224 citations

On Hull-Variation Problem of Equivalent Linear Codes

Hao Chen

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.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.

arXiv:2206.14516Jun 29, 20220 citations

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.