Related papers: Five-Lee-weight linear codes over $\mathbb{F}_{q}+…
Let $p$ be a prime number and $\varsigma$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^\varsigma$ over $\mathcal{R}$ are precisely the…
Generalized quasi-cyclic (GQC) codes with arbitrary lengths over the ring $\mathbb{F}_{q}+u\mathbb{F}_{q}$, where $u^2=0$, $q=p^n$, $n$ a positive integer and $p$ a prime number, are investigated. By the Chinese Remainder Theorem,…
In this paper, a family of five-weight reducible cyclic codes is presented. Furthermore, the weight distribution of these cyclic codes is determined, which follows from the determination of value distributions of certain exponential sums.
Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do…
In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ using…
Projective two-weight linear codes are closely related to finite projective spaces and strongly regular graphs. In this paper, a family of $q$-ary projective two-weight linear codes is presented, where $q$ is a power of 2. The parameters of…
We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding…
Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…
The weight hierarchy of a linear code has been an important research topic in coding theory since Wei's original work in 1991. Choosing $ D=\Big\{(x,y)\in \Big(\F_{p^{s_1}}\times\F_{p^{s_2}}\Big)\Big\backslash\{(0,0)\}: f(x)+g(y)=0\Big\}$…
Subfield codes of linear codes over finite fields have recently received much attention. Some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the $q$-ary…
Linear codes with few weights have been a significant area of research in coding theory for many years, due to their applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. Inspired by…
For which positive integers $n,k,r$ does there exist a linear $[n,k]$ code $C$ over $\mathbb{F}_q$ with all codeword weights divisible by $q^r$ and such that the columns of a generating matrix of $C$ are projectively distinct? The…
We study sequences of linear or affine codes with uniform weight spectrum, i.e., a part of codewords with any fixed weight tends to zero. It is proved that a sequence of linear codes has a uniform weight spectrum if the number of vectors…
In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary…
In this article we present a class of codes with few weights arising from special type of linear sets. We explicitly show the weights of such codes, their weight enumerator and possible choices for their generator matrices. In particular,…
We investigate additive cyclic codes over the alphabet $\mathbb{F}_{q}\mathbb{F}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of $\mathbb{F}_{q^2}$-additive…
In the recent work \cite{shi18}, a combinatorial problem concerning linear codes over a finite field $\F_q$ was introduced. In that work the authors studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero distinct…
In this article we mainly study linear codes over $\mathbb{F}_{2^n}$ and their binary subfield codes. We construct linear codes over $\mathbb{F}_{2^n}$ whose defining sets are the certain subsets of $\mathbb{F}_{2^n}^m$ obtained from…
We develop an algorithm for computing the weight distribution of a linear $[n,k]$ code over a finite field $\mathbb{F}_q$. We represent the codes by their characteristic vector with respect to a given generator matrix and a special type of…
Few-weight codes over finite chain rings are associated with combinatorial objects such as strongly regular graphs (SRGs), strongly walk-regular graphs (SWRGs) and finite geometries, and are also widely used in data storage systems and…