Related papers: Construction and enumeration of left dihedral code…
In the field of algebraic geometric codes (AG codes), the characterization of dual codes has long been a challenging problem which relies on differentials. In this paper, we provide some descriptions for certain differentials utilizing…
The hull of a linear code is defined as the intersection of the code and its dual. This concept was initially introduced to classify finite projective planes. The hull plays a crucial role in determining the complexity of algorithms used to…
We define the Euclidean hull of a linear code $C$ as the intersection of $C$ and its Euclidean dual $C^\perp$. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing…
Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$, $R=\mathbb{F}_{2^m}+u\mathbb{F}_{2^m}$ $(u^2=0)$ and $s,n$ be positive integers such that $n$ is odd. In this paper, we give an explicit representation for every self-dual…
The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right…
Split group codes are a class of group algebra codes over an abelian group. They were introduced in 2000 by Ding, Kohel and Ling as a generalization of the cyclic duadic codes. For a prime power q and an abelian group G of order n such that…
Duadic codes are a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. For every prime power q, we characterize the integers n such that over the finite field with q^2 elements there is a duadic…
Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault…
Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length $n$ over a finite field if and only if $n$ and the field characteristic are even. The enumeration of such…
Algorithms to construct minimal left group codes are provided. These are based on results describing a complete set of orthogonal primitive idempotents in each Wedderburn component of a semisimple finite group algebra FG for a large class…
In this article, we introduce new scalar products over finite rings via additive isomorphisms. This allows us to define new notions of right (respectively left) orthogonal codes, that are not necessarily linear. This leads to definitions of…
Let $\mathbb{F}_{2^m}$ be a finite field of cardinality $2^m$ and $s$ a positive integer. Using properties for Kronecker product of matrices and calculation for linear equations over $\mathbb{F}_{2^m}$, an efficient method for the…
In this article, we study the derivations of group algebras of some important groups, namely, dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($SD_{8n}$). First, we explicitly classify all inner derivations of a group algebra…
From a given $[n, k]$ code $C$, we give a method for constructing many $[n, k]$ codes $C'$ such that the hull dimensions of $C$ and $C'$ are identical. This method can be applied to constructions of both self-dual codes and linear…
Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$, $R=\mathbb{F}_{q}[u]/\langle u^4\rangle=\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}+u^3\mathbb{F}_{q}$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer…
The Euclidean hull of a linear code $C$ is defined as $C\cap C^{\perp}$, where $C^\perp$ denotes the dual of $C$ under the Euclidean inner product. A linear code with zero hull dimension is called a linear complementary dual (LCD) code. A…
The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new…
In this paper, we investigate the existence of self-dual MRD codes $C\subset L^n$, where $L/F$ is an arbitrary field extension of degree $m\geq n$. We then apply our results to the case of finite fields, and prove that if $m=n$ and…
Bazzi and Mitter [3] showed that binary dihedral group codes are asymptotically good. In this paper we prove that the dihedral group codes over any finite field with good mathematical properties are asymptotically good. If the…
In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the direct sum construction and the $(\mathbf{u},\mathbf{u+v})$-construction. Some new criteria for the…