Constacyclic codes over finite commutative semi-simple rings

Finite commutative semi-simple rings are direct sum of finite fields. In this study, we investigate the algebraic structure of λ-constacyclic codes over such finite semi-simple rings. Among others, necessary and sufficient conditions for the existence of self-dual, LCD, and Hermitian dual-containing λ-constacyclic codes over finite semi-simple rings are provided. Using the CSS and Hermitian constructions, quantum MDS codes over finite semi-simple rings are constructed.     

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) are of much interest from many viewpoints due to their theoretical and practical properties. 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 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.     

Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.     

New entanglement-assisted quantum codes from k-Galois dual codes

Entanglement-assisted quantum error-correcting (EAQEC, for short) codes use pre-existing entanglements between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, they can only be constructed from classical linear codes which contain their Hermitian dual codes. However, how to determine the parameters of ebits c in EAQEC codes is not an easy task. In this paper, let p be prime and e, k be integers, we construct six classes of EAQEC codes based on k-Galois dual codes over finite fields ${\mathbb{F}}_{p^e}$, where $0\le k<e$. The parameter of ebits c of these EAQEC codes can be easily generated algebraically. Furthermore, the six classes of EAQEC codes are of maximal entanglement, most of which have better parameters than current EAQEC codes available.     

Galois self-dual extended duadic constacyclic codes

Galois inner product is a generalization of the Euclidean inner product and Hermitian inner product. The theory on linear codes under Galois inner product can be applied in the constructions of MDS codes and quantum error-correcting codes. In this paper, we construct Galois self-dual codes and MDS Galois self-dual codes from extensions of constacyclic codes. First, we explicitly determine all the Type II splittings leading to all the Type II duadic constacyclic codes in two cases. Second, we propose methods to extend two classes of constacyclic codes to obtain Galois self-dual codes, and we also provide existence conditions of Galois self-dual codes which are extensions of constacyclic codes. Finally, we construct some (almost) MDS Galois self-dual codes using the above results. Some Galois self-dual codes and (almost) MDS Galois self-dual codes obtained in this paper turn out to be new.     

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.     

Construction of binary LCD codes,ternary LCD codes and quaternary Hermitian LCD codes

We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bounds on the largest minimum weights.

     

A new concatenated type construction for LCD codes and isometry codes

We give a new concatenated type construction for linear codes with complementary dual (LCD) over small finite fields. In this construction,we need a special class of inner codes that we call isometry codes. Our construction generalizes a recent construction of Carlet et al. (2014–2016) and of Güneri et al. (2016). In particular, it allows us to construct LCD codes with improved parameters directly.     

Constructions of several classes of linear codes with a few weights

Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over ${\mathbb{F}}_p$, where $p$ is an odd prime. The weight distributions of these constructed codes are also settled by applications of the theory of quadratic forms and Gauss sums over finite fields. Some of the linear codes obtained are optimal or almost optimal. The parameters of these linear codes are new in most cases. Moreover, two classes of MDS codes are obtained.     

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

Nearly Kähler and Hermitian <Emphasis Type="Italic">f</Emphasis>-structures on homogeneous Φ-spaces of order <Emphasis Type="Italic">k</Emphasis> with special metrics

We consider arbitrary homogeneous Φ-spaces of order k ≥ 3 of semisimple compact Lie groups G in the case of a series of special metrics. We give formulas for the Nomizu function of the Levi-Civita connection of these metrics. Using these formulas and other relations for Φ-spaces of order k, we prove necessary and sufficient conditions for the canonical f-structures on these spaces to lie in some generalized Hermitian geometry classes of f-structures: nearly Kähler (NKf-structures) and Hermitian (Hf-structures).     

Self-dual Co des with Symplectic Inner Pro duct

In this paper, we discuss a kind of Hermitian inner product — symplectic inner product, which is different from the original inner product — Euclidean inner product. According to the definition of symplectic inner product, the codes under the symplectic inner product have better properties than those under the general Hermitian inner product. Here we present the necessary and sufficient condition for judging whether a linear code C over Fp with a generator matrix in the standard form is a symplectic self-dual code. In addition, we give a method for constructing a new symplectic self-dual codes over Fp, which is simpler than others.     

Constructions of maximum distance separable symbol-pair codes using cyclic and constacyclic codes

Symbol-pair code is a new coding framework which is proposed to correct errors in the symbol-pair read channel. In particular, maximum distance separable (MDS) symbol-pair codes are a kind of symbol-pair codes with the best possible error-correction capability. Employing cyclic and constacyclic codes, we construct three new classes of MDS symbol-pair codes with minimum pair-distance five or six. Moreover, we find a necessary and sufficient condition which ensures a class of cyclic codes to be MDS symbol-pair codes. This condition is related to certain property of a special kind of linear fractional transformations. A detailed analysis on these linear fractional transformations leads to an algorithm, which produces many MDS symbol-pair codes with minimum pair-distance seven.     

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.     

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.