A Note on The Enumeration of Euclidean Self-Dual Skew-Cyclic Codes over Finite Fields

In this paper we give the enumeration formulas for Euclidean self-dual skew-cyclic codes over finite fields when $(n,|\theta|)=1$ and for some cases when $(n,|\theta|)>1,$ where $n$ is the length of the code and $|\theta|$ is the order of automorphism $\theta.

Skew-constacyclic codes over Fq[v]⟨ vq−v ⟩\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}

In this paper, the investigation on the algebraic structure of the ring $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given

On Codes over Fq+vFq+v2Fq\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring R=\F_{q}+v\F_{q}+v^{2}\F_{q}, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over \F_q and extend these to codes over $R$.Comment: 19 page

Skew-Polynomial Rings and Skew-Cyclic Codes

This is a survey on the theory of skew-cyclic codes based on skew-polynomial rings of automorphism type. Skew-polynomial rings have been introduced and discussed by Ore (1933). Evaluation of skew polynomials and sets of (right) roots were first considered by Lam (1986) and studied in great detail by Lam and Leroy thereafter. After a detailed presentation of the most relevant properties of skew polynomials, we survey the algebraic theory of skew-cyclic codes as introduced by Boucher and Ulmer (2007) and studied by many authors thereafter. A crucial role will be played by skew-circulant matrices. Finally, skew-cyclic codes with designed minimum distance are discussed, and we report on two different kinds of skew-BCH codes, which were designed in 2014 and later.Comment: This survey will appear as a chapter in "A Concise Encyclopedia of Coding Theory" to be published by CRC Pres

Skew Constacyclic Codes over Finite Chain Rings

Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings ([8]) have been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Of more interest are codes over the ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. The structure of all skew constacyclic codes is completely determined. This allows us to express generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length~2 over $\mathbb{F}_{3}+u\mathbb{F}_{3}$ and their Euclidean and Hermitian dual codes is also provided.Comment: 24 Pages, Submitted to Advances in Mathematics of Communication

On skew polynomial codes and lattices from quotients of cyclic division algebras

We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak p\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak p$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak p\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.Comment: 15 page

A note on the duals of skew constacyclic codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $\Phi_{\alpha,\theta}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $\Phi_{\alpha,\theta}(a_0,...,a_{n-2}, a_{n-1}):=(\alpha \theta(a_{n-1}),\theta(a_0),...,\theta(a_{n-2}))$ for some $\alpha\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.Comment: 31 pages, 3 tables; this is a revised version that includes improvements to the presentation of the main results, a new subsection and an appendix which is an extension of Section 2 of the previous versio

Dual codes of product semi-linear codes

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with linear codes of $\mathbb{F}_q^n$ invariant under a semi-linear map $T:\mathbb{F}_q^n\to\mathbb{F}_q^n$ for some $n\geq 2$. In particular, we study three kind of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring $\mathbb{F}_q[X,\theta]$ as a subcase of linear codes invariant by a semi-linear map $T$. In this setting we give also an algorithm for encoding, decoding and detecting errors and we show a method to construct codes invariant under a fixed $T$.Comment: v1: 37 pages; v2: 31 pages, the presentation of the main topics is improved by some minor changes and additional results, and some mistakes and typos have been correcte

Structure of linear codes over the ring BkB_k

We study the structure of linear codes over the ring $B_k$ which is defined by $\mathbb{F}_{p^r}[v_1,v_2,\ldots,v_k]/\langle v_i^2=v_i,~v_iv_j=v_jv_i \rangle_{i,j=1}^k.$ In order to study the codes, we begin with studying the structure of the ring $B_k$ via a Gray map which also induces a relation between codes over $B_k$ and codes over $\mathbb{F}_{p^r}.$ We consider Euclidean and Hermitian self-dual codes, MacWilliams relations, as well as Singleton-type bounds for these codes. Further, we characterize cyclic and quasi-cyclic codes using their images under the Gray map, and give the generators for these type of codes.Comment: 18 pages, accepted for publication (Journal of Applied Mathematics and Computing

Polyadic cyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv−vu⟩\mathbb{F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle

Let $f(u)$ and $g(v)$ be any two polynomials of degree $k$ and $\ell$ respectively ($k$ and $\ell$ are not both $1$), which split into distinct linear factors over $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u,v]/\langle f(u),g(v),uv-vu\rangle$ be a finite commutative non-chain ring. In this paper, we study polyadic codes and their extensions over the ring $\mathcal{R}$. We give examples of some polyadic codes which are optimal with respect to Griesmer type bound for rings. A Gray map is defined from $\mathcal{R}^n \rightarrow \mathbb{F}^{k\ell n}_q$ which preserves duality. The Gray images of polyadic codes and their extensions over the ring $\mathcal{R}$ lead to construction of self-dual, isodual, self-orthogonal and complementary dual (LCD) codes over $\mathbb{F}_q$. Some examples are also given to illustrate this