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In this paper, we determine the parameters of $\mathbb{Z}_q$-MacDonald Code of dimension k for any positive integer $q \geq 2.$ Further, we have obtained the weight distribution of $\mathbb{Z}_q$-MacDonald code of dimension 3 and…
We study the generalized rank weight distribution of a linear code. First, we provide a MacWilliams-type identity which relates the distributions of a code and its dual. Then, we give a formula for the enumerator polynomial. Finally, we…
Let $\mathbb{F}_q$ denote the finite field of order $q,$ $n$ be a positive integer coprime to $q$ and $t \geq 2$ be an integer. In this paper, we enumerate all the complementary-dual cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes of…
Relations between the local weight distributions of a binary linear code, its extended code, and its even weight subcode are presented. In particular, for a code of which the extended code is transitive invariant and contains only codewords…
We introduce the concept of spread of a code, and we specialize it to the case of maximum weight spectrum (MWS) codes. We classify two newly-defined sub-families of MWS codes according to their weight distributions, and completely describe…
Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of $q$-ary linear codes derived from irreducible cyclic…
In this paper, we propose a class of linear codes and obtain their weight distribution. Some of these codes are almost optimal. Moreover, several classes of constant composition codes(CCCs) are constructed as subcodes of linear codes.
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring…
We compute the weight distribution of the ${\mathcal R} (4,9)$ by combining the approach described in D. V. Sarwate's Ph.D. thesis from 1973 with knowledge on the affine equivalence classification of Boolean functions. To solve this problem…
In this paper, we first introduce the notion of generalized pair weights of an $[n, k]$-linear code over the finite field $\mathbb{F}_q$ and the notion of pair $r$-equiweight codes, where $1\le r\le k-1$. Some basic properties of…
The $b$-symbol read channel is motivated by the limitations of the reading process in high density data storage systems. The corresponding new metric is a generalization of the Hamming metric known as the $b$-symbol weight metric and has…
Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4).$ Let $\mathbb{F}_{p^m}$ be the finite field with $p^m$ elements, and let $R=\mathbb{F}_{p^m}[u]/\left<u^2\right>$ be the finite commutative…
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…
Due to their efficient encoding and decoding algorithms cyclic codes, a subclass of linear codes, have applications in consumer electronics, data storage systems, and communication systems. In this paper, Dickson polynomials of the first…
We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies…
In this paper, we completely determine the number of solutions to $ \operatorname{Tr}^{q^2}_q(bx+b)+c=0, x\in \mu_{q+1}\backslash \{-1\}$ for all $b\in \mathbb{F}_{q^2}, c\in\mathbb{F}_{q}$. As an application, we can give the weight…
In this paper, we derive the average weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rate of the average weight distribution in the limit of…
We study the generalized and extended weight enumerator of the q-ary Simplex code and the q-ary first order Reed-Muller code. For our calculations we use that these codes correspond to a projective system containing all the points in a…
Linear codes with a few weights are very important in coding theory and have attracted a lot of attention. In this paper, we present a construction of $q$-ary linear codes from trace and norm functions over finite fields. The weight…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for a prime $p$, we determine the explicit complete weight enumerators of a family of linear codes over $\mathbb{F}_p$ with defining set…