Related papers: Valued rank-metric codes
Scattered polynomials of a given index over finite fields are intriguing rare objects with many connections within mathematics. Of particular interest are the exceptional ones, as defined in 2018 by the first author and Zhou, for which…
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…
We study the rank of complex sparse matrices in which the supports of different columns have small intersections. The rank of these matrices, called design matrices, was the focus of a recent work by Barak et. al. (BDWY11) in which they…
We initiate a study of varieties of minimal degree in weighted projective spaces. We call a weighted projective space $\mathbf{P}(w_0,\dots,w_n)$ divisible if $w_i \mid w_{i+1}$ for all $i$. We provide sharp bounds for when a non-degenerate…
The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic…
The aim of this note (as well as of the course itself) is to give a largely self-contained proof of two of the main results in the field of low-rank matrix recovery. This field aims for identification of low-rank matrices from only limited…
Design matrices are sparse matrices in which the supports of different columns intersect in a few positions. Such matrices come up naturally when studying problems involving point sets with many collinear triples. In this work we consider…
In this paper, we propose a metric on the space of finite sets of trajectories for assessing multi-target tracking algorithms in a mathematically sound way. The main use of the metric is to compare estimates of trajectories from different…
There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed…
Critical loci for projective reconstruction from three views in four dimensional projective space are defined by an ideal generated by maximal minors of suitable $4 \times 3$ matrices, $N,$ of linear forms. Such loci are classified in this…
Rank-metric codes were studied by E. Gabidulin in 1985 after a brief introduction by Delsarte in 1978 as an equivalent of Reed-Solomon codes, but based on linearized polynomials. They have found applications in many areas, including linear…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…
In rank-metric cryptography, a vector from a finite dimensional linear space over a finite field is viewed as the linear space spanned by its entries. The rank decoding problem which is the analogue of the problem of decoding a random…
We construct an explicit family of linear rank-metric codes over any field ${\mathbb F}_h$ that enables efficient list decoding up to a fraction $\rho$ of errors in the rank metric with a rate of $1-\rho-\epsilon$, for any desired $\rho \in…
After recalling the definition of codes as modules over skew polynomial rings, whose multiplication is defined by using an automorphism and a derivation, and some basic facts about them, in the first part of this paper we study some of…
This paper introduces new constructions of sum-rank metric codes derived from algebraic function fields, as existing results on such codes remain limited. A major challenge lies in the determination of their parameters. We address this…
In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $\mathbb{P}^n(\mathbb{F}_q)$, and they may be seen…
In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted…
In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic…
Error-correcting codes have an important role in data storage and transmission and in cryptography, particularly in the post-quantum era. Hermitian matrices over finite fields and equipped with the rank metric have the potential to offer…