Michel Lavrauw, John Sheekey
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256
本文通过引入新不变量和计算技术，完全分类了包含16阶核的256阶半域。
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.
In [U. Dempwolff: More Translation Planes and Semifields from Dembowski-Ostrom Polynomials, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81-103], the author gives a construction of three classes of rank two semifields of order $q^{2n}$, with $q$ and $n$ odd, using Dembowski-Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for $n>3$, in particular we prove that two of these classes, labeled $D_{A}$ and $D_{AB}$, are new for $n>3$, whereas presemifields in family …
In [U. Dempwolff: \textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103] three classes of rank two presemifields of order $q^{2n}$, with $q$ and $n$ odd, were exhibited, leaving as an open problem the isotopy issue. In [M. Lavrauw, G. Marino, O. Polverino, R. Trombetti: \textit{Solution to an isotopism question concerning rank 2 semifields}, J. Comb. Des., to appear. DOI: 10.1002/jcd.21382], the authors faced with this problem answering the question whether these presemifields are new for $n>3$. In this pa…
The BEL-construction for finite semifields was introduced in \cite{BEL2007}; a geometric method for constructing semifield spreads, using so-called BEL-configurations in $V(rn,q)$. In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in $V(rn,q)$, extending the results from \cite{BEL2007}, where this was obtained only for $r=n$. Given a BEL-configuration with associated semifields spread $\mathcal{S}$, we also show how to find a BEL-configuration corresponding to the dual spread $\mathca…
On Translation Hyperovals in Semifield Planes
本文构造了第一个不含平移超卵形线的有限平移平面（广义扭曲域平面，阶64），否定了Cherowitzo猜想，并将其与秩度量码覆盖半径及散射子空间不存在性相关联。
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 …
In this paper we study a family of scattered $\F_q$--linear sets of rank $tn$ of the projective space $PG(2n-1,q^t)$ ($n \geq 1$, $t\geq 3$), called of {\it pseudoregulus type}, generalizing results contained in [G. Marino, O. Polverino, R. Trombetti: On ${\mathbb F}_q$--linear sets of $\PG(3,q^3)$ and semifields, {\em J. Combin. Theory Ser. A}, {\bf 114} (2007), 769--788] and in [M. Lavrauw, G. Van de Voorde: Scattered linear sets and pseudoreguli, preprint]. As an application, we characterize, in terms of the associated linear sets, some classical families of semifields: the Generalized Twis…
On the autotopism groups and the equivalence of finite cyclic semifields
本文给出有限循环半域同伦的完全分类及自同构群的完全确定，解决了长期未决问题，还对通过斜多项式构造的最大秩距离码进行分类并描述其自同构群，推进了相关领域研究。
Rank metric codes from Drinfeld modules
本文通过Drinfeld模与秩度量码的联系，构建了基于线性子空间的秩度量码，并给出了新的半字段码构造方法。
In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,…
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…
An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。