Related papers: Weight distributions of cyclic codes with respect …

We study the functional codes $C_2(X)$ defined on a projective variety $X$, in the case where $X \subset {\mathbb{P}}^3$ is a non-degenerate hermitian surface. We first give some bounds for $# X_{Z(\mathcal{Q})}(\mathbb{F}_{q})$, which are…

Let R > S be finite Frobenius rings for which there exists a trace map T from R onto S as left S modules. Let C:= {x -> T(ax + bf(x)) : a,b in R}. Then C is an S-linear subring-subcode of a left linear code over R. We consider functions f…

We study trace codes with defining set $L,$ a subgroup of the multiplicative group of an extension of degree $m$ of the alphabet ring $\mathbb{F}_3+u\mathbb{F}_3+u^{2}\mathbb{F}_{3},$ with $u^{3}=1.$ These codes are abelian, and their…

In this paper, based on the theory of defining sets, two classes of at most six-weight linear codes over $\mathbb{F}_p$ are constructed. The weight distributions of the linear codes are determined by means of Gaussian period and Weil sums.…

BCH codes are an important class of linear codes and find extensive utilization in communication and disk storage systems.This paper mainly analyzes the negacyclic BCH code and cyclic BCH code of length $\frac{q^m-1}{2}$. For negacyclic BCH…

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 $F_p$ be the prime field with $p$ elements. We derive the homogeneous weight on the Frobenius matrix ring $M_2(F_p)$ in terms of the generating character. We also give a generalization of the Lee weight on the finite chain ring…

The weight distribution and weight hierarchy of a linear code are two important research topics in coding theory. In this paper, choosing $ D=\Big\{(x,y)\in \Big(\F_{p^{s_1}}\times\F_{p^{s_2}}\Big)\Big\backslash\{(0,0)\}:…

We derive a recursive formula determing the weight distribution of the [n=(q^m-1)/(q-1), n-m, 3] Hamming code H(m,q), when (m, q-1)=1. Here q is a prime power. The proof is based on Moisio's idea of using Pless power moment identity…

In this paper, we construct an infinite family of three-weight binary codes from linear codes over the ring $R=\mathbb{F}_2+v\mathbb{F}_2+v^2\mathbb{F}_2$, where $v^3=1.$ These codes are defined as trace codes. They have the algebraic…

In this paper, based on the theory of defining sets, two classes of five-weight or six-weight linear codes over Fp are constructed. The weight distributions of the linear codes are determined by means of Weil sums and a new type of…

Let $R=\mathbb{Z}_{4}[u]/\langle u^k\rangle=\mathbb{Z}_{4}+u\mathbb{Z}_{4}+\ldots+u^{k-1}\mathbb{Z}_{4}$ ($u^k=0$) where $k\in \mathbb{Z}^{+}$ satisfies $k\geq 2$. For any odd positive integer $n$, it is known that cyclic codes over $R$ of…

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…

Combinatorial $t$-designs have wide applications in coding theory, cryptography, communications and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a $t$-design. In this paper, we first…

The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d <q$ and $d>(n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not…

Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a $t$-design. Till now only a small amount of work on constructing $t$-designs…

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…

Recently, $b$-symbol codes are proposed to protect against $b$-symbol errors in $b$-symbol read channels. It is an interesting subject of study to consider the complete $b$-symbol weight distribution of cyclic codes since $b$-symbol metric…

Let $p$ be a prime and $\mathbb{F}_q$ be the finite field of order $q=p^m$. In this paper, we study $\mathbb{F}_q\mathcal{R}$-skew cyclic codes where $\mathcal{R}=\mathbb{F}_q+u\mathbb{F}_q$ with $u^2=u$. To characterize…

As a generalization of cyclic codes of length p^s over F_{p^a}, we study n-dimensional cyclic codes of length p^{s_1} X ... X p^{s_n} over F_{p^a} generated by a single "monomial". Namely, we study multi-variable cyclic codes of the form…