Related papers: On maximum additive Hermitian rank-metric codes

Motivated by classification, up to order isomorphism, of some dense subgroups of Euclidean space that are free of minimal rank, we obtain apparently new invariants for an equivalence relation (intermediate between Hermite and Smith) on…

Linear codes with small hulls over finite fields have been extensively studied due to their practical applications in computational complexity and information protection. In this paper, we develop a general method to determine the exact…

Recent work shows that a new family of norms on Hermitian matrices arise by evaluating the even degree complete homogeneous symmetric (CHS) polynomials on the eigenvalues of a Hermitian matrix. The CHS norm of a graph is then defined by…

Duadic codes are a class of cyclic codes that generalizes quadratic residue codes from prime to composite lengths. For every prime power q, we characterize the integers n such that over the finite field with q^2 elements there is a duadic…

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…

We propose the Legendrian web in a contact three manifold as a second order generalization of the planar web. An Abelian relation for a Legendrian web is analogously defined as an additive equation among the first integrals of its…

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call \emph{visible rank}. The locality constraints of a linear code are…

The group of combinatorial self-similarities of a pseudometric space $(X, d)$ is the maximal subgroup of the symmetric group $\mathbf{Sym} (X)$ whose elements preserve the four-point equality $d(x,y)=d(u,v)$. Let us denote by $\mathcal{IP}$…

Let $\mathcal{C}$ be a set of $m$ by $n$ matrices over $\mathbb{F}_q$ such that the rank of $A-B$ is at least $d$ for all distinct $A,B\in \mathcal{C}$. Suppose that $m\leqslant n$. If $\#\mathcal{C}= q^{n(m-d+1)}$, then $\mathcal{C}$ is a…

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to…

Rank similarity measures provide a method for quantifying differences between search engine results without the need for relevance judgments. For example, the providers of a search service might use such measures to estimate the impact of a…

In this work, we study linear codes with the folded Hamming distance, or equivalently, codes with the classical Hamming distance that are linear over a subfield. This includes additive codes. We study MDS codes in this setting and define…

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over $\F_{4}$ are used in the construction of many asymmetric quantum codes…

We study the locally recoverable codes on algebraic curves. In the first part of this article, we provide a bound of generalized Hamming weight of these codes. Whereas in the second part, we propose a new family of algebraic geometric LRC…

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…

A sum-rank-metric code attaining the Singleton bound is called maximum sum-rank distance (MSRD). MSRD codes have been constructed for some parameter cases. In this paper we construct a linear MSRD code over an arbitrary field ${\bf F}_q$…

The $\Z_{p^s}$-additive codes of length $n$ are subgroups of $\Z_{p^s}^n$, and can be seen as a generalization of linear codes over $\Z_2$, $\Z_4$, or $\Z_{2^s}$ in general. A $\Z_{p^s}$-linear generalized Hadamard (GH) code is a GH code…

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…

With a grading previously introduced by the second-named author, the multiplication maps in the preprojective algebra satisfy a maximal rank property that is similar to the maximal rank property proven by Hochster and Laksov for the…

Let $q$ be an odd prime power and let $X(m,q)$ be the set of symmetric bilinear forms on an $m$-dimensional vector space over $\mathbb{F}_q$. The partition of $X(m,q)$ induced by the action of the general linear group gives rise to a…