Related papers: Anticodes in the Sum-Rank Metric
Motivated by applications to the theory of rank-metric codes, we study the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection…
We investigate two fundamental questions intersecting coding theory and combinatorial geometry, with emphasis on their connections. These are the problem of computing the asymptotic density of MRD codes in the rank metric, and the Critical…
We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over…
This paper provides new bounds on the size of spheres in any coordinate-additive metric with a particular focus on improving existing bounds in the sum-rank metric. We derive improved upper and lower bounds based on the entropy of a…
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…
This paper investigates the construction of rank-metric codes with specified Ferrers diagram shapes. These codes play a role in the multilevel construction for subspace codes. A conjecture from 2009 provides an upper bound for the dimension…
We consider the decoding of rank metric codes assuming the error matrix is symmetric. We prove two results. First, for rates $<1/2$ there exists a broad family of rank metric codes for which any symmetric error pattern, even of maximal rank…
Codes in the sum-rank metric have received many attentions in recent years, since they have wide applications in the multishot network coding, the space-time coding and the distributed storage. In this paper, by constructing covering codes…
This is a chapter of the upcoming "A Concise Encyclopedia of Coding Theory", W.C. Huffman, J.-L. Kim, and P. Sole' Eds., CRC Press. The chapter gives an introduction to the mathematical theory of rank-metric codes. Treated topics include:…
Finding the maximum size of a Sidon set in $\mathbb{F}_2^t$ is of research interest for more than 40 years. In order to tackle this problem we recall a one-to-one correspondence between sum-free Sidon sets and linear codes with minimum…
In this paper we give new constructions of Ferrer diagram rank metric codes, which achieve the largest possible dimension. In particular, we prove several cases of a conjecture by T. Etzion and N. Silberstein. We also establish a sharp…
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 study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening…
Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with…
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.).…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of…
We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte…
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right…
Lattices and partially ordered sets have played an increasingly important role in coding theory, providing combinatorial frameworks for studying structural and algebraic properties of error-correcting codes. Motivated by recent works…
In this paper, we study the congruences, prime filters and prime ideals of horizontal sums of bounded lattices, then, through a construction based on horizontal sums and without enforcing the Continuum Hypothesis, we are modifying an…