Related papers: Consta-dihedral Codes over Finite Fields

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…

This is a manuscript of a chapter prepared for a book. The good codes possess large information length and large minimum distance. A class of codes is said to be asymptotically good if there exists a positive real $\delta$ such that, for…

In this paper, we investigate the existence and asymptotic property of self-dual $2$-quasi negacyclic codes of length $2n$ over a finite field of cardinality $q$. When $n$ is odd, we show that the $q$-ary self-dual $2$-quasi negacyclic…

Let $F$ be a field with cardinality $p^\ell$ and $0\neq \lambda\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we…

A kind of self-dual quasi-abelian codes of index $2$ over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes of this kind whose relative minimum weights are small, such codes are proved to…

In this paper, we show that LCD codes are not equivalent to linear codes over small finite fields. The enumeration of binary optimal LCD codes is obtained. We also get the exact value of LD$(n,2)$ over $\mathbb{F}_3$ and $\mathbb{F}_4$. We…

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…

Many kinds of codes which possess two cycle structures over two special finite commutative chain rings, such as ${\Bbb Z}_2{\Bbb Z}_4$-additive cyclic codes and quasi-cyclic codes of fractional index etc., were proved asymptotically good.…

Multi-dimensional cyclic code is a natural generalization of cyclic code. In an earlier paper we explored two-dimensional constacyclic codes over finite fields. Following the same technique, here we characterize the algebraic structure of…

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…

In this paper we characterize the algebraic structure of two-dimensional $(\alpha,\beta )$-constacyclic codes of arbitrary length $s.\ell$ and of their duals. For $\alpha,\beta \in \{1,-1\}$, we give necessary and sufficient conditions for…

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good…

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=\F_{q}+v\F_{q}+v^{2}\F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and…

The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence.…

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we…

In this paper, we study some repeated-root two-dimensional cyclic and constacyclic codes over a finite field $F=\mathbb{F}_q$. We obtain the generator matrices and generator polynomials of these codes and their duals. We also investigate…

We consider codes over the two semi-local non-unital rings of order six, \[ H_{23} = \langle a,b \mid 2a=0, 3b = 0, a^2=a, b^2 = 0, ab = 0 = ba \rangle,\] and \[H_{32} = \langle a,b \mid 2a=0, 3b = 0, a^2=0, b^2 = b, ab = 0 = ba \rangle. \]…

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains…

We study classical and quantum LDPC codes of constant rate obtained by the lifted product construction over non-abelian groups. We show that the obtained families of quantum LDPC codes are asymptotically good, which proves the qLDPC…

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following ideas of Bazzi and Mitter on group codes over the binary field, we prove that group codes over finite fields of any characteristic are…