Construction of some new quantum MDS codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper, we mainly apply a new method of classical Hermitian self-orthogonal codes to construct three classes of new quantum MDS codes, and these quantum MDS codes provide large minimum distance.     

Some new quantum MDS codes from generalized Reed–Solomon codes

Quantum maximum distance separable (MDS) codes form a significant class of quantum codes. In this paper, by using Hermitian self-orthogonal generalized Reed–Solomon codes, we construct two new classes of $q$-ary quantum MDS codes, which have minimum distance greater than $\frac{q}{2}$. Most of these quantum MDS codes are new in the sense that their parameters are not covered by the codes available in the literature.     

Quantum MDS codes with new length and large minimum distance

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes $GR{S}_{k_1}(A,{\mathit{v}}_A)$ and $GR{S}_{k_2}(B,{\mathit{v}}_B)$, we propose a sufficient condition to ensure that $GR{S}_k(A\cup B,{\mathit{v}}_{A\cup B})$ is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q-ary quantum MDS codes are bigger than $q/2+1$.     

New quantum MDS codes with large minimum distance and short length from generalized Reed–Solomon codes

Quantum maximum-distance-separable (MDS) codes are an important class of quantum codes. In this paper we mainly use classical Hermitian self-orthogonal generalized Reed–Solomon codes to construct three classes of new quantum MDS codes. Further, these quantum MDS codes have large minimum distance and short length.     

Constructing self-orthogonal and Hermitian self-orthogonal codes via weighing matrices and orbit matrices

We define the notion of an orbit matrix with respect to standard weighing matrices, and with respect to types of weighing matrices with entries in a finite field. In the latter case we primarily restrict our attention the fields of order 2, 3 and 4. We construct self-orthogonal and Hermitian self-orthogonal linear codes over finite fields from these types of weighing matrices and their orbit matrices respectively. We demonstrate that this approach applies to several combinatorial structures such as Hadamard matrices and balanced generalized weighing matrices. As a case study we construct self-orthogonal codes from some weighing matrices belonging to some well known infinite families, such as the Paley conference matrices, and weighing matrices constructed from ternary periodic Golay pairs.     

On the constructions of MDS self-dual codes via cyclotomy

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers.     

Construction of long MDS self-dual codes from short codes

In this paper, we propose a mechanism on how to construct long MDS self-dual codes from short ones. These codes are special types of generalized Reed-Solomon (GRS) codes or extended generalized Reed-Solomon codes. The main tool is utilizing additive structure or multiplicative structure on finite fields. By applying this method, more MDS self-dual codes can be constructed.     

Construction of MDS Self-dual Codes over Finite Fields

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For finite field with odd characteristic and square cardinality, our results can produce more classes of MDS self-dual codes than previous works.     

On Hermitian LCD codes and their Gray image

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.     

Structure of functional codes defined on non-degenerate Hermitian varieties

We study the functional codes of order h defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by Edoukou (2009) [8]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first 2h+1 weights.     

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over ${\mathbb{F}}_{q^m}$ are GRS codes (resp., extended GRS codes) when $m=1$. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.     

The asymptotic probability distribution of the relative distance of additive quantum codes

For given information rate R, it is proved as n tends to infinite, that almost all additive ?n,n⋅R? quantum codes (pure and impure) are the codes with their relative distance tending to h⁻¹(1−R), where is an entropy function.