Polyadic Constacyclic Codes

For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier of a Type I polyadic constacyclic code is given. As an application, some optimal codes from Type I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant MDS codes, are constructed.Comment: We provide complete solutions on two basic questions on polyadic constacyclic cdes, and construct some optimal codes from the polyadic constacyclic cde

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

mm-adic residue codes over Fq[v]/(vs−v)\mathbb{F}_q[v]/(v^s-v)

Due to their rich algebraic structure, cyclic codes have a great deal of significance amongst linear codes. Duadic codes are the generalization of the quadratic residue codes, a special case of cyclic codes. The $m$-adic residue codes are the generalization of the duadic codes. The aim of this paper is to study the structure of the $m$-adic residue codes over the quotient ring $\mathbb{F}_{q}[v]/(v^s-v).$ We determine the idempotent generators of the $m$-adic residue codes over $\mathbb{F}_{q}[v]/(v^s-v)$. We obtain some parameters of optimal $m$-adic residue codes over $\mathbb{F}_{q}[v]/(v^s-v),$ with respect to Griesmer bound for rings

Computational Results of Duadic Double Circulant Codes

Quadratic residue codes have been one of the most important classes of algebraic codes. They have been generalized into duadic codes and quadratic double circulant codes. In this paper we introduce a new subclass of double circulant codes, called {\em{duadic double circulant codes}}, which is a generalization of quadratic double circulant codes for prime lengths. This class generates optimal self-dual codes, optimal linear codes, and linear codes with the best known parameters in a systematic way. We describe a method to construct duadic double circulant codes using 4-cyclotomic cosets and give certain duadic double circulant codes over $\mathbb F_2, \mathbb F_3, \mathbb F_4, \mathbb F_5$, and $\mathbb F_7$. In particular, we find a new ternary self-dual $[76,38,18]$ code and easily rediscover optimal binary self-dual codes with parameters $[66,33,12]$, $[68,34,12]$, $[86,43,16]$, and $[88,44,16]$ as well as a formally self-dual binary $[82,41,14]$ code.Comment: 12 pages, 5 tabels, to appear in J. of Applied Mathematics and Computin

Skew constacyclic 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 two polynomials of degree $k$ and $\ell$ respectively, not both linear, 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 $\psi$-skew cyclic and $\theta_t$-skew constacyclic codes over the ring $\mathcal{R}$ where $\psi$ and $\theta_t$ are two automorphisms defined on $\mathcal{R}$.Comment: 15 page

Iso-Orthogonality and Type II Duadic Constacyclic Codes

Generalizing even-like duadic cyclic codes and Type-II duadic negacyclic codes, we introduce even-like (i.e.,Type-II) and odd-like duadic constacyclic codes, and study their properties and existence. We show that even-like duadic constacyclic codes are isometrically orthogonal, and the duals of even-like duadic constacyclic codes are odd-like duadic constacyclic codes. We exhibit necessary and sufficient conditions for the existence of even-like duadic constacyclic codes. A class of even-like duadic constacyclic codes which are alternant MDS-codes is constructed

Duadic negacyclic codes over a finite non-chain ring and their Gray images

Let $f(u)$ be a polynomial of degree $m, m \geq 2,$ which splits into distinct linear factors over a finite field $\mathbb{F}_{q}$. Let $\mathcal{R}=\mathbb{F}_{q}[u]/\langle f(u)\rangle$ be a finite non-chain ring. In an earlier paper, we studied duadic and triadic codes over $\mathcal{R}$ and their Gray images. Here, we study duadic negacyclic codes of Type I and Type II over the ring $\mathcal{R}$, their extensions and their Gray images. As a consequence some self-dual, isodual, self-orthogonal and complementary dual(LCD) codes over $\mathbb{F}_q$ are constructed. Some examples are also given to illustrate this.Comment: arXiv admin note: text overlap with arXiv:1609.0786

Constacyclic symbol-pair codes: lower bounds and optimal constructions

Symbol-pair codes introduced by Cassuto and Blaum (2010) are designed to protect against pair errors in symbol-pair read channels. The higher the minimum pair distance, the more pair errors the code can correct. MDS symbol-pair codes are optimal in the sense that pair distance cannot be improved for given length and code size. The contribution of this paper is twofold. First we present three lower bounds for the minimum pair distance of constacyclic codes, the first two of which generalize the previously known results due to Cassuto and Blaum (2011) and Kai {\it et al.} (2015). The third one exhibits a lower bound for the minimum pair distance of repeated-root cyclic codes. Second we obtain new MDS symbol-pair codes with minimum pair distance seven and eight through repeated-root cyclic codes

Knots as processes: a new kind of invariant

We exhibit an encoding of knots into processes in the {\pi}-calculus such that knots are ambient isotopic if and only their encodings are weakly bisimilar

Constacyclic and Quasi-Twisted Hermitian Self-Dual Codes over Finite Fields

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing $x^n-\lambda$ over $\mathbb{F}_{q^2}$ is given, where $\lambda$ is a unit in $\mathbb{F}_{q^2}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda$-constacyclic codes of length $n$ over $\mathbb{F}_{q^2}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over $\mathbb{F}_{q^2}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$ is introduced. As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda$ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length some over extension fields of $\mathbb{F}_{q^2}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well