Symplectic self-orthogonal quasi-cyclic codes

In this paper, we obtain sufficient and necessary conditions for quasi-cyclic codes with index even to be symplectic self-orthogonal. Then, we propose a method for constructing symplectic self-orthogonal quasi-cyclic codes, which allows arbitrary polynomials that coprime $x^{n}-1$ to construct symplectic self-orthogonal codes. Moreover, by decomposing the space of quasi-cyclic codes, we provide lower and upper bounds on the minimum symplectic distances of a class of 1-generator quasi-cyclic codes and their symplectic dual codes. Finally, we construct many binary symplectic self-orthogonal codes with excellent parameters, corresponding to 117 record-breaking quantum codes, improving Grassl's table (Bounds on the Minimum Distance of Quantum Codes. http://www.codetables.de)

On the Structure of the Linear Codes with a Given Automorphism

The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual

Quasi-cyclic Hermitian construction of binary quantum codes

In this paper, we propose a sufficient condition for a family of 2-generator self-orthogonal quasi-cyclic codes with respect to Hermitian inner product. Supported in the Hermitian construction, we show algebraic constructions of good quantum codes. 30 new binary quantum codes with good parameters improving the best-known lower bounds on minimum distance in Grassl's code tables \cite{Grassl:codetables} are constructed

Generator polynomial matrices of the Galois hulls of multi-twisted codes

In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field $\mathbb{F}_{p^e}$ of characteristic $p$. Let $\mathbf{G}$ be a generator polynomial matrix (GPM) of a MT code $\mathcal{C}$. For any $0\le \kappa<e$, the $\kappa$-Galois hull of $\mathcal{C}$, denoted by $h_\kappa\left(\mathcal{C}\right)$, is the intersection of $\mathcal{C}$ with its $\kappa$-Galois dual. The main result in this paper is that a GPM for $h_\kappa\left(\mathcal{C}\right)$ has been obtained from $\mathbf{G}$. We start by associating a linear code $\mathcal{Q}_\mathbf{G}$ with $\mathbf{G}$. We show that $\mathcal{Q}_\mathbf{G}$ is quasi-cyclic. In addition, we prove that the dimension of $h_\kappa\left(\mathcal{C}\right)$ is the difference between the dimension of $\mathcal{C}$ and that of $\mathcal{Q}_\mathbf{G}$. Thus the determinantal divisors are used to derive a formula for the dimension of $h_\kappa\left(\mathcal{C}\right)$. Finally, we deduce a GPM formula for $h_\kappa\left(\mathcal{C}\right)$. In particular, we handle the cases of $\kappa$-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing optimal and maximum distance separable codes, are used to illustrate the theoretical results

A Class of Quantum LDPC Codes Constructed From Finite Geometries

Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self-orthogonal with containing only one cycle of length four. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, the encoding and decoding algorithms of these codes as well as their performance over various quantum depolarizing channels will be investigated.Comment: 5pages, 2 figure

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) have been of much interest from many researchers due to their theoretical significant and practical implications. However, little work has been done on LCD MDS codes. In particular, determining the existence of $q$-ary $[n,k]$ LCD MDS codes for various lengths $n$ and dimensions $k$ is a basic and interesting problem. In this paper, we firstly study the problem of the existence of $q$-ary $[n,k]$ LCD MDS codes and completely solve it for the Euclidean case. More specifically, we show that for $q>3$ there exists a $q$-ary $[n,k]$ Euclidean LCD MDS code, where $0\le k \le n\le q+1$, or, $q=2^{m}$, $n=q+2$ and $k= 3 \text{or} q-1$. Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes