A geometric characterisation of Desarguesian spreads

We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.     

Combinatorial invariants for nets of conics in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )

Designs, Codes and Cryptography - The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics...     

Rank metric codes and zeta functions

We define the rank metric zeta function of a code as a generating function of its normalized q-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.     

The BEL-rank of finite semifields

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.     

Rank properties of subspaces of symmetric and hermitian matrices over finite fields

We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank.