Related papers: Primitive Idempotents and Constacyclic Codes over …
Let $R$ be the finite chain ring $\mathbb{F}_{p^{2m}}+{u}\mathbb{F}_{p^{2m}}$, where $\mathbb{F}_{p^{2m}}$ is the finite field with $p^{2m}$ elements, $p$ is a prime, $m$ is a non-negative integer and ${u}^{2}=0.$ In this paper, we firstly…
Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the…
Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime number and $m$ is a positive integer. Self-dual constacyclic codes of length \( p^s \) over \( \frac{\mathbb{F}_{p^m}[u]}{\langle u^3 \rangle} \) exist only…
An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents…
In this paper we investigate the class of constacyclic codes, which is a natural generalization of the class of cyclic and negacyclic codes. This class of codes is interesting in the sense that it contains codes with good or even optimal…
A complete set of primitive orthogonal idempotents plays an important role in the representation theory of an associative algebra. In this paper, we construct a complete set of primitive orthogonal idempotents for any finite Brandt…
Properties of matrix product codes over finite commutative Frobenius rings are investigated. The minimum distance of matrix product codes constructed with several types of matrices is bounded in different ways. The duals of matrix product…
We investigate Buchbaum and Eisenbud's construction of the second symmetric power S^2_R(X) of a chain complex X of modules over a commutative ring R. We state and prove a number of results from the folklore of the subject for which we know…
Let R be a connected noetherian commutative ring, and let G be a simply connected reductive group over R of isotropic rank ge 2. The elementary subgroup E(R) of G(R) is the subgroup generated by the R-points U_P^+(R) and U_P^-(R) of the…
Let $\mathbb{F}_{p^{m}}$ be a finite field with cardinality $p^{m}$ and $R=\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$ with $u^{2}=0$. We aim to determine all $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where…
The product matrix of a finite commutative ring $R=\{x_1,x_2,\ldots,x_n\}$ and an element $u \in R$ is the matrix $A_u(R)=[a_{ij}]$, where $a_{ij}=1$ if $x_ix_j=u$, and $a_{ij}=0$ otherwise. This provides a natural extension of the concept…
As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus…
In this paper, a transform approach is used for polycyclic and serial codes over finite local rings in the case that the defining polynomials have no multiple roots. This allows us to study them in terms of linear algebra and invariant…
In this paper, we study the linear codes over the commutative ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$ and their Gray images, where $v^{3}=v$. We define the Lee weight of the elements of $R$, we give a Gray map from $R^{n}$ to $\F^{3n}_{q}$ and…
It is well-known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group $G$ can be detected on the cohomology group $\mathrm{H}^1(G,R[G])$, where $R$ is either a finite field, the…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of…
We give an elementary proof prove of the preservation of the Noetherian condition for commutative rings with unity $R$ having at least one finitely generated ideal $I$ such that the quotient ring is again finitely generated, and $R$ is…
We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x;\theta] modulo the left ideal generated by a polynomial of the type x^n-a. This allows us to develop an approach to skew-constacyclic codes based on…
We describe an algorithm for computing the complete set of primitive orthogonal idempotents in the centralizer ring of the permutation representation of a wreath product. This set of idempotents determines the decomposition of the…