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In this work we develop a geometric approach to the study of rank metric codes. Using this method, we introduce a simpler definition for generalized rank weight of linear codes. We give a complete classification of constant rank weight code…
The problem of finding the maximal dimension of linear or affine subspaces of matrices whose rank is constant, or bounded below, or bounded above, has attracted many mathematicians from the sixties to the present day. The problem has caught…
In this paper, we first introduce the concept of elementary linear subspace, which has similar properties to those of a set of coordinates. We then use elementary linear subspaces to derive properties of maximum rank distance (MRD) codes…
In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…
The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are…
Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight…
We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector…
The sum-rank metric can be seen as a generalization of both, the rank and the Hamming metric. It is well known that sum-rank metric codes outperform rank metric codes in terms of the required field size to construct maximum distance…
In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved…
New lower bounds on the minimum average Hamming distance of binary codes are derived. The bounds are obtained using linear programming approach.
In this paper we study geometric aspects of codes in the sum-rank metric. We establish the geometric description of generalised weights, and analyse the Delsarte and geometric dual operations. We establish a correspondence between maximum…
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.
The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with…
Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
The correspondence between linear codes and representable matroids is well known. But a similar correspondence between quantum codes and matroids is not known. We show that representable symplectic matroids over a finite field…
Let $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ be an $\mathbb{F}_{q^m}$-linear non-degenerate rank metric code with dimension $k$. In this paper we investigate the problem of determining the number $M(\mathcal{C})$ of codewords in…
Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of $t$-weight linear codes over ${\mathbb F}_{q}$ are presented with…
Two skew cyclic codes can be equivalent for the Hamming metric only if they have the same length, and only the zero code is degenerate. The situation is completely different for the rank metric, where lengths of codes correspond to the…
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,…