相关论文: Whitney Numbers of Rank-Metric Lattices and Code E…
We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We…
We present the theory of linear rank-metric codes from the point of view of their fundamental parameters. These are: the minimum rank distance, the rank distribution, the maximum rank, the covering radius, and the field size. The focus of…
We study the Whitney numbers of the first kind of combinatorial geometries. The first part of the paper is devoted to general results relating the M\"{o}bius functions of nested atomistic lattices, extending some classical theorems in…
This paper investigates general properties of codes with the rank metric. We first investigate asymptotic packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their…
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:…
This work investigates the structure of rank-metric codes in connection with concepts from finite geometry, most notably the $q$-analogues of projective systems and blocking sets. We also illustrate how to associate a classical…
We investigate the asymptotic density of error-correcting codes with good distance properties and prescribed linearity degree, including sublinear and nonlinear codes. We focus on the general setting of finite translation-invariant metric…
This study investigates Hermitian rank-metric codes, a special class of rank-metric codes, focusing on perfect codes and on the analysis of their covering properties. Firstly, we establish bounds on the size of spheres in the space of…
We review the main results of the theory of rank-metric codes, with emphasis on their combinatorial properties. We study their duality theory and MacWilliams identities, comparing in particular rank-metric codes in vector and matrix…
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…
In this paper we introduce and investigate rank-metric intersecting codes, a new class of linear codes in the rank-metric context, inspired by the well-studied notion of intersecting codes in the Hamming metric. A rank-metric code is said…
This paper investigates packing and covering properties of codes with the rank metric. First, we investigate packing properties of rank metric codes. Then, we study sphere covering properties of rank metric codes, derive bounds on their…
We introduce the class of partition-balanced families of codes, and show how to exploit their combinatorial invariants to obtain upper and lower bounds on the number of codes that have a prescribed property. In particular, we derive precise…
Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by $T$, producing an asymptotic estimate as $T \to \infty$. This problem can be…
Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number…
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…
It is shown that the Whitney function of a representable q-matroid and the collection of all higher weight enumerators of any representing rank-metric code determine each other via a monomial substitution. Moreover, the q-matroid itself and…
A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error…
We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions…
We introduce a new technique to construct rank-metric codes using the arithmetic theory of Drinfeld modules over global fields, and Dirichlet Theorem on polynomial arithmetic progressions. Using our methods, we obtain a new infinite family…