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 Problem for combinatorial geometries by Crapo and Rota. Using methods from semifield theory, we derive two lower bounds for the density function of full-rank, square MRD codes. The first bound is sharp when the matrix size is a prime number and the underlying field is sufficiently large, while the second bound applies to the binary field. We then take a new look at the Critical Problem for combinatorial geometries, approaching it from a qualitative, often asymptotic, viewpoint. We illustrate the connection between this very classical problem and that of computing the asymptotic density of MRD codes. Finally, we study the asymptotic density of some special families of codes in the rank metric, including the symmetric, alternating and Hermitian ones. In particular, we show that the optimal codes in these three contexts are sparse.
In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields.
We determine the tensor rank of all semifields of order 16 over $\mathbb{F}_2$ and of all semifields of order 81 over $\mathbb{F}_3$. Our results imply that some semifields of order 81 have lower multiplicative complexity than the finite field $\mathbb{F}_{81}$ over $\mathbb{F}_3$. We prove new results on the correspondence between linear codes and tensor rank, including a generalisation of a theorem of Brockett and Dobkin to arbitrary tensors, which makes the problem computationally feasible.
We classify the rank two commutative semifields which are 8-dimensional over their center $\mathbb{F}_{q}$. This is done using computational methods utilizing the connection to linear sets in $\mathrm{PG}(2,q^{4})$. We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over $\mathbb{F}_{3}$. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.
The isotopism problem of a class of 6-dimensional rank 2 semifields and its solution
在美国，Dempwolff：\textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}，设计，代码，密码\textbf{68}（1-3）（2013），81- 103]展示了三类秩为$q^{2n}$的二阶前半域，其中$q$和$n$为奇数，留下了同向问题作为开放问题。在[M.拉夫劳湾马里诺岛波尔韦里诺河Trombetti：\textit{Solution to an isotopism question concerning rank 2 semifields}，J. Comb. Des.，出现。DOI：10.1002/jcd.21382]，作者面对这个问题，回答这些前半字段是否是$n>3$的新字段。在本文中，我们完成了研究解决的情况下$n=3$。
In this article we construct a new family of semifields, containing and extending two well-known families, namely Albert's generalised twisted fields and Petit's cyclic semifields (also known as Johnson-Jha semifields). The construction also gives examples of semifields with parameters for which no examples were previously known. In the case of semifields two dimensions over a nucleus and four-dimensional over their centre, the construction gives all possible examples. Furthermore we embed these semifields in a new family of maximum rank-distance codes, encompassing most known current construc…
In this paper we prove a result on the structure of the elements of an additive {\it maximum rank distance (MRD) code} over the field of order two, namely that in some cases such codes must contain a semifield spread set. We use this result to classify additive MRD codes in $M_n(\mathbb{F}_2)$ with minimum distance $n-1$ for $n\leq 6$. Furthermore we present a computational classification of additive MRD codes in $M_4(\mathbb{F}_3)$. The computational evidence indicates that MRD codes of minimum distance $n-1$ are much more rare than MRD codes of minimum distance $n$, i.e. semifield spread set…
In this paper, we present a new family of maximum rank distance (MRD for short) codes in $\mathbb F_{q}^{2n\times 2n}$ of minimum distance $2\leq d\leq 2n$. In particular, when $d=2n$, we can show that the corresponding semifield is exactly a Hughes-Kleinfeld semifield. The middle and right nuclei of these MRD codes are both equal to $\mathbb F_{q^n}$. We also prove that the MRD codes of minimum distance $2<d<2n$ in this family are inequivalent to all known ones. The equivalence between any two members of this new family is also determined.
We determine the proportion of $[3\times 3;3]$-MRD codes over ${\mathbb F}_q$ within the space of all $3$-dimensional $3\times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $q\rightarrow\infty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.
Maximum scattered linear sets of pseudoregulus type and the Segre Variety ${\cal S}_{n,n}$
本文研究了射影空间$PG(2n-1,q^t)$（$n \geq 1$，$t\geq 3$）的一类离散$\F_q$ --秩为$tn$的线性集，称为{ \it伪正则型}，推广了[G.马里诺岛波尔韦里诺河Trombetti：On ${\mathbb F}_q$ --线性集合的$\PG(3,q^3)$和半域，{ \em J. Combin. Theory Ser. A}，{ \bf 114}（2007），769- 788]和[M.拉夫劳湾Van de Voorde：Scattered linear sets and pseudoreguli，preprint].作为应用，我们用相关的线性集刻画了一些经典的半域族：广义扭域和二维Knuth半域。
Université de Rennes（雷恩大学）；Université Paris 8, Laboratoire de Géométrie, Analyse et Applications, LAGA, Université Sorbonne Paris Nord, CNRS（巴黎第八大学，几何、分析与应用实验室，拉格朗日分析几何实验室，索邦巴黎北大学，法国国家科学研究中心）；Department of Mathematics and Applications “R. Caccioppoli”, University of Naples Federico II（那不勒斯费德里科二世大学数学与应用“R.卡奇奥波利”系）
In this article we construct a new family of linear maximum rank distance (MRD) codes for all parameters. This family contains the only known family for general parameters, the Gabidulin codes, and contains codes inequivalent to the Gabidulin codes. This family also contains the well-known family of semifields known as Generalised Twisted Fields. We also calculate the automorphism group of these codes, including the automorphism group of the Gabidulin codes.
This preprint is of a chapter to appear in {\it Combinatorics and finite fields: Difference sets, polynomials, pseudorandomness and applications. Radon Series on Computational and Applied Mathematics}, K.-U. Schmidt and A. Winterhof (eds.). Rank-metric codes are codes consisting of matrices with entries in a finite field, with the distance between two matrices being the rank of their difference. Codes with maximum size for a fixed minimum distance are called Maximum Rank Distance (MRD) codes. Such codes were constructed and studied independently by Delsarte (1978), Gabidulin (1985), Roth (1991…
The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line $\mathrm{PG}(1,q^n)$. In this paper, we provide a large family of new maximum scattered linear sets over $\mathrm{PG}(1,q^n)$ for any even $n\geq 6$ and odd $q$. In particular, the relevant family contains …
The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. T…
We investigate the rank of the average mixing matrix of trees, with all eigenvalues distinct. The rank of the average mixing matrix of a tree on $n$ vertices with $n$ distinct eigenvalues is upper-bounded by $\frac{n}{2}$. Computations on trees up to $20$ vertices suggest that the rank attains this upper bound most of the times. We give an infinite family of trees whose average mixing matrices have ranks which are bounded away from this upper bound. We also give a lower bound on the rank of the average mixing matrix of a tree.
We find a simple, closed formula for the proportion of vertices which are $k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find that, as $n$ goes to infinity, the average rank of a random vertex in a tree of size $n$ approaches 0.727649, and the average rank of the root of a tree of size $n$ approaches 1.62297.
