Related papers: Composite Matrices from Group Rings, Composite G-C…
This paper has two main parts. In the first part we develop an elementary coordinatization for any nilpotent group $G$ taking exponents in a binomial principal ideal domain (PID) $A$. In case that the additive group $A^+$ of $A$ is finitely…
Let $p$ be a prime such that $p \equiv 2$ or $3$ mod $5$. Linear block codes over the non-commutative matrix ring of $2 \times 2$ matrices over the prime field $GF(p)$ endowed with the Bachoc weight are derived as isometric images of linear…
We consider additive codes over GF(4) that are self-dual with respect to the Hermitian trace inner product. Such codes have a well-known interpretation as quantum codes and correspond to isotropic systems. It has also been shown that these…
In this paper we construct binary self-dual codes using the \'etale cohomology of $\mathbb{Z}/2$ on the spectra of rings of $S$-integers of global fields. We will show that up to equivalence, all self-dual codes of length at least 4 arise…
A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over…
We construct two new infinite families of trace codes of dimension $2m$, over the ring $\mathbb{F}_p+u\mathbb{F}_p,$ when $p$ is an odd prime. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by…
Binary constant weight codes have important applications and have been studied for many years. Optimal or near-optimal binary constant weight codes of small lengths have been determined. In this paper we propose a new construction of…
A random field $X = (X_v)_{v \in G}$ on a quasi-transitive graph $G$ is a factor of i.i.d. if it can be written as $X=\varphi(Y)$ for some i.i.d. process $Y= (Y_v)_{v \in G}$ and equivariant map $\varphi$. Such a map, also called a coding,…
We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with…
We construct a family of two-Lee-weight codes over the ring $R_k,$ which is defined as trace codes with algebraic structure of abelian codes. The Lee weight distribution of the two-weight codes is given. Taking the Gray map, we obtain…
An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional…
Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as QC-LDPC codes, have many applications due to their simple encoding implementation by means of cyclic shift registers. In this paper, we construct QC-LDPC codes from…
In classical coding theory, Gray isometries are usually defined as mappings between finite Frobenius rings, which include the ring $Z_m$ of integers modulo $m$, and the finite fields. In this paper, we derive an isometric mapping from $Z_8$…
In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we…
Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators,…
As a generalization of cyclic codes, quasi-cyclic (QC) codes contain many good linear codes. But quasi-cyclic codes studied so far are mainly limited to one generator (1-generator) QC codes. In this correspondence, 2-generator and…
Properties of a given symmetry group G are very important in investigation of a physical system invariant under its action. In the case of finite spin systems (magnetic rings, some planar macromolecules) the symmetry group is isomorphic…
When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic…
The results of J. F. Qiann et al. [4] on $(1-\gamma)$-cyclic codes over finite chain rings of nilpotency index 2 are extended to $(1-\gamma^e)$-cyclic codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map is…
In the present paper, we discuss the class of Type III and Type IV codes from the perspectives of neighbors. Our investigation analogously extends the results originally presented by Dougherty [8] concerning the neighbor graph of binary…