Related papers: Rank-Metric Codes and $q$-Polymatroids
The ranks and kernels of generalized Hadamard matrices are studied. It is proven that any generalized Hadamard matrix $H(q,\lambda)$ over $F_q$, $q>3$, or $q=3$ and $\gcd(3,\lambda)\not =1$, generates a self-orthogonal code. This result…
We describe a construction of the Tutte polynomial for both matroids and $q$-matroids based on an appropriate partition of the underlying support lattice into intervals that correspond to prime-free minors, which we call a Tutte partition.…
A $t$-$(n,d,\lambda)$ design over ${\mathbb F}_q$, or a subspace design, is a collection of $d$-dimensional subspaces of ${\mathbb F}_q^n$, called blocks, with the property that every $t$-dimensional subspace of ${\mathbb F}_q^n$ is…
In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary…
This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…
Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association…
Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix…
We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed…
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work.…
In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one…
We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary…
The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely…
In this paper, we generalize the method of using two parallel versions of the lifted MRD code from the existing work [1]. The Delsarte theorem of the rank distribution of MRD codes is an important part to count codewords in our…
We introduce ideas that complement the many known connections between polymatroids and graph coloring. Given a hypergraph that satisfies certain conditions, we construct polymatroids, given as rank functions, that can be written as sums of…
Skew polynomial rings provide a fundamental example of noncommutative principal ideal domains. Special quotients of these rings yield matrix algebras that play a central role in the theory of rank-metric codes. Recent breakthroughs have…
In this paper we present several natural $q$-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some relating analytical results and ask for combinatorial interpretations.
Latroids were introduced by Vertigan, who associated a latroid to a linear block code and showed that its Tutte polynomial determines the weight enumerator of the code. We associate a latroid to a code over a ring or a field endowed with a…
Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of…
The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an $\mathbb{F}_q$-linear rank-metric code over $\mathbb{F}_q^{m \times n}$ of rate $R =…