Related papers: Spread Codes from Abelian non-cyclic groups

In this work we construct a new class of maximal partial spreads in $PG(4,q)$, that we call $q$-added maximal partial spreads. We obtain them by depriving a spread of a hyperplane of some lines and adding $q+1$ lines not of the hyperplane…

We propose a new method of constructing q-ary propelinear perfect codes. The approach utilizes permutations of the fixed length q-ary vectors that arise from the automorphisms of the regular subgroups of the affine group. For any prime q it…

For each odd prime power $q$, let $4 \leq n\leq q^{2}+1$. Hermitian self-orthogonal $[n,2,n-1]$ codes over $GF(q^{2})$ with dual distance three are constructed by using finite field theory. Hence, $[[n,n-4,3]]_{q}$ quantum MDS codes for $4…

In this paper we will study cyclic codes over some special rings: F_{q}[u]/(u^{i}), F_{q}[u_1,...u_{i}]/(u_1^2,u_2^2,...,u_{i}^2, u_1 u_2 - u_2 u_1,...,u_{i}u_{j} - u_{j}u_{i},...), F_{q}[u,v]/(u^{i},v^{j},uv-vu), q=p^{r}, where p is a…

Let $q$ be a power of a prime $p$. In this paper, we study reversible cyclic codes of arbitrary length over the ring $ R = \mathbb{F}_q + u \mathbb{F}_q$, where $u^2=0 mod q$. First, we find a unique set of generators for cyclic codes over…

Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let $R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with $u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of $(\theta, \Theta)$-cyclic…

It has been shown that good structured codes over non-Abelian groups do exist. Specifically, we construct codes over the smallest non-Abelian group $\mathds{D}_6$ and show that the performance of these codes is superior to the performance…

The distribution of the number of points on abelian covers of $\mathbb{P}1(\mathbb{F}_q)$ ranging over an irreducible moduli space has been answered in recent work by the author. Bucur, et al. determined the distribution over the whole…

In this paper, we define Abelian and consta-Abelian polyadic codes over rings defined as affine algebras over chain rings. For that aim, we use the classical construction via splittings and multipliers of the underlying Abelian group. We…

{A cyclic subspace code is a union of the orbits of subspaces contained in it. In a recent paper, Gluesing-Luerssen et al. (Des. Codes Cryptogr. 89, 447-470, 2021) showed that the study of the distance distribution of a single orbit cyclic…

Let $q$ be a power of a prime $p$, $G$ be a finite abelian group, where $p$ does not divide $|G|$,and let $n$ be a positive integer. In this paper we find a formula for the number of irreducible representations of $G$ of a given dimension…

Let $\mathbb {F}_q$ be a finite field and $G$ a finte group with $(|G|,q)=1$. By a group code in $\mathbb {F}_q[G]$ we mean a two-sided ideal in $\mathbb {F}_q[G]$. We will prove a general criterion for the existence of group codes with…

The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner…

Generalising a previous result, we determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding $10^4$. The computer code for this and similar calculations is made available.

For any positive integer $m$ and an odd prime $p$; let $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $q=p^{m}$, be a ring extension of the ring $\mathbb{F}_{p}+u\mathbb{F}_{p}.$ In this paper, we construct linear codes over…

Let $p$ be a prime number, $q=p^s$ for a positive integer $s$. For any positive divisor $e$ of $q-1$, we construct an infinite family codes of size $q^{2m}$ with few Lee-weight. These codes are defined as trace codes over the ring…

For a fixed dimension $N$ we compute the generating function of the numbers $t_N(n)$ (respectively $\bar{t}_N(n)$) of $PGL_{N+1}(k)$-orbits of rational $n$-sets (respectively rational $n$-multisets) of the projective space $\mathb{P}^N$…

We prove that a set $A$ of at most $q$ non-collinear points in the finite plane $\mathbb{F}_{q}^{2}$ spans at least $\approx\frac{|A|}{\sqrt{q}}$ directions: this is based on a lower bound contained in [FST13], which we prove again together…

We will describe an algorithm to construct an elliptic curve $E_{f_q}$ over some prime field $\mathbb{F}_p$ such that such that $|E_{f_q}(\mathbb{F}_p)| = f_q$, where $f_q$ is a probable Fibonacci prime for some prime index $q$. The…

Let $D$ be a division algebra, finite-dimensional over its center, and $R=D[t;\sigma,\delta]$ a skew polynomial ring. Using skew polynomials $f\in R$, we construct division algebras and a generalization of maximum rank distance codes…