Related papers: Non-minimum tensor rank Gabidulin codes
This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also…
In 2019, Trombetti and Zhou introduced a new family of $\mathbb{F}_{q^n}$-linear Maximum Rank Distance (MRD) codes over $\mathbb{F}_{q^{2n}}$. For such codes we propose a new syndrome-based decoding algorithm. It is well known that a…
The problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of K\"otter and Kschischang. A large class of constant-dimension subspace codes is…
We study the rank weight hierarchy, thus in particular the rank metric, of cyclic codes over the finite field $\mathbb F_{q^m}$, $q$ a prime power, $m \geq 2$. We establish the rank weight hierarchy for $[n,n-1]$ cyclic codes and…
There exist linear relations among tensor entries of low rank tensors. These linear relations can be expressed by multi-linear polynomials, which are called generating polynomials. We use generating polynomials to compute tensor rank…
We generalise Gabidulin codes to the case of infinite fields, eventually with characteristic zero. For this purpose, we consider an abstract field extension and any automorphism in the Galois group. We derive some conditions on the…
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric.…
It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a questioned original posed in a paper on a taxonomy of 2-primal rings. Answers to similar questions on minimal rings relating to this taxonomy…
After a seminal paper by Shekeey (2016), a connection between maximum $h$-scattered $\mathbb{F}_q$-subspaces of $V(r,q^n)$ and maximum rank distance (MRD) codes has been established in the extremal cases $h=1$ and $h=r-1$. In this paper, we…
We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs…
Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent…
In this paper, we mainly study quaternary linear codes and their binary subfield codes. First we obtain a general explicit relationship between quaternary linear codes and their binary subfield codes in terms of generator matrices and…
The minimum rank of a graph is the minimum possible rank of a real matrix whose zero-nonzero pattern is described by the graph. The current algorithms can compute efficiently the minimum rank of undirected trees. This paper provides an…
The minimum rank of a simple graph $G$ is defined to be the smallest possible rank over all symmetric real matrices whose $ij$th entry (for $i\neq j$) is nonzero whenever $\{i,j\}$ is an edge in $G$ and is zero otherwise. Minimum rank is a…
Optimal rank-metric codes in Ferrers diagrams can be used to construct good subspace codes. Such codes consist of matrices having zeros at certain fixed positions. This paper generalizes the known constructions for Ferrers diagram…
We transpose the theory of rank metric and Gabidulin codes to the case of fields which are not finite fields. The Frobenius automorphism is replaced by any element of the Galois group of a cyclic algebraic extension of a base field. We use…
We prove a lower bound on the dimension of the set of maximal border subrank tensors. This is the first such bound of its type.
We give a self-contained classification of $1_*$-generic minimal border rank tensors in $\mathbb{C}^m \otimes \mathbb{C}^m \otimes \mathbb{C}^m$ for $m \leq 5$. Together with previous results, this gives a classification of all minimal…
The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult;…
We give a general lower bound on the rank of matrices of the form $\rho(h) - I$ with $\rho : G \rightarrow GL({\mathbb F}^n)$ an irreducible representation of a finite group $G$. The main tool in the proof is a (strengthening) of a…