Code–checkable group rings

A code over a group ring is defined to be a submodule of that group ring. For a code $C$ over a group ring $RG$, $C$ is said to be checkable if there is $v\in RG$ such that {$C=\{x\in RG: xv=0\}$}. In \cite{r2}, Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring $RG$ is code-checkable if every ideal in $RG$ is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring $\mathbb{F}G$, when $\mathbb{F}$ is a finite field and $G$ is a finite abelian group, to be code-checkable. In this paper, we give some characterizations for code-checkable group rings for more general alphabet. For instance, a finite commutative group ring $RG$, with $R$ is semisimple, is code-checkable if and only if $G$ is $\pi'$-by-cyclic $\pi$; where $\pi$ is the set of noninvertible primes in $R$. Also, under suitable conditions, $RG$ turns out to be code-checkable if and only if it is pseudo-morphic

Dihedral codes with prescribed minimum distance

Dihedral codes, particular cases of quasi-cyclic codes, have a nice algebraic structure which allows to store them efficiently. In this paper, we investigate it and prove some lower bounds on their dimension and minimum distance, in analogy with the theory of BCH codes. This allows us to construct dihedral codes with prescribed minimum distance. In the binary case, we present some examples of optimal dihedral codes obtained by this construction.Comment: 13 page

NonCommutative Rings and their Applications, IV ABSTRACTS Checkable Codes from Group Algebras to Group Rings

Abstract A code over a group ring is defined to be a submodule of that group ring. For a code C over a group ring RG, C is said to be checkable if there is v ∈ RG such that C = {x ∈ RG : xv = 0}. In [1], Jitman et al. introduced the notion of code-checkable group ring. We say that a group ring RG is code-checkable if every ideal in RG is a checkable code. In their paper, Jitman et al. gave a necessary and sufficient condition for the group ring FG, when F is a finite field and G is a finite abelian group, to be codecheckable. In this paper, we generalize this result for RG, when R is a finite commutative semisimple ring and G is any finite group. Our main result states that: Given a finite commutative semisimple ring R and a finite group G, the group ring RG is code-checkable if and only if G is π -by-cyclic π; where π is the set of noninvertible primes in R

CSS-like Constructions of Asymmetric Quantum Codes

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of $q$-ary AQCs is extended by removing the \F_{q}-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure $q$-ary CSS-like codes for $q \in {2,3,4,5,7,8,9}$ up to reasonable lengths are found. In many instances, removing the \F_{q}-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.Comment: Accepted by IEEE Trans. Information Theory in June 2013, to appea

Contemporary Coding Theory

Coding Theory naturally lies at the intersection of a large number of disciplines in pure and applied mathematics. A multitude of methods and means has been designed to construct, analyze, and decode the resulting codes for communication. This has suggested to bring together researchers in a variety of disciplines within Mathematics, Computer Science, and Electrical Engineering, in order to cross-fertilize generation of new ideas and force global advancement of the field. Areas to be covered are Network Coding, Subspace Designs, General Algebraic Coding Theory, Distributed Storage and Private Information Retrieval (PIR), as well as Code-Based Cryptography

Dimensión de ideales en álgebras de grupo, y códigos de grupo

En este trabajo se determinan algunas cotas y relaciones para la dimensión de ideales principales en álgebras de grupo analizando polinomios mínimos de representaciones regulares. Estos resultados son utilizados, primero, en el contexto de álgebras de grupo semisimples, para calcular, para cualquier código abeliano, un elemento con peso de Hamming igual a su dimensión. Luego, para obtener cotas de la distancia mínima de ciertos códigos MDS. Una relación entre una clase de códigos de grupo y códigos MDS es presentada. Se exponen ejemplos ilustrando los resultados principales