Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields

Subgeometries and linear sets on a projective line

We define the splash of a subgeometry on a projective line, extending the definition of \cite{BaJa13} to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank $r$, generalizing the definition from \cite{FaSz2006}, and show that clubs correspond to tangent splashes. We determine the condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent

On the equivalence of linear sets

Let $L$ be a linear set of pseudoregulus type in a line $\ell$ in $\Sigma^*=\mathrm{PG}(t-1,q^t)$, $t=5$ or $t>6$. We provide examples of $q$-order canonical subgeometries $\Sigma_1,\, \Sigma_2 \subset \Sigma^*$ such that there is a $(t-3)$-space $\Gamma \subset \Sigma^*\setminus (\Sigma_1 \cup \Sigma_2 \cup \ell)$ with the property that for $i=1,2$, $L$ is the projection of $\Sigma_i$ from center $\Gamma$ and there exists no collineation $\phi$ of $\Sigma^*$ such that $\Gamma^{\phi}=\Gamma$ and $\Sigma_1^{\phi}=\Sigma_2$. Condition (ii) given in Theorem 3 in Lavrauw and Van de Voorde (Des. Codes Cryptogr. 56:89-104, 2010) states the existence of a collineation between the projecting configurations (each of them consisting of a center and a subgeometry), which give rise by means of projections to two linear sets. It follows from our examples that this condition is not necessary for the equivalence of two linear sets as stated there. We characterize the linear sets for which the condition above is actually necessary.Comment: Preprint version. Referees' suggestions and corrections implemented. The final version is to appear in Designs, Codes and Cryptograph

Non-linear maximum rank distance codes in the cyclic model for the field reduction of finite geometries

In this paper we construct infinite families of non-linear maximum rank distance codes by using the setting of bilinear forms of a finite vector space. We also give a geometric description of such codes by using the cyclic model for the field reduction of finite geometries and we show that these families contain the non-linear maximum rank distance codes recently provided by Cossidente, Marino and Pavese.Comment: submitted; 22 page

Translation hyperovals and F-2-linear sets of pseudoregulus type

In this paper, we study translation hyperovals in PG(2, q(k)). The main result of this paper characterises the point sets defined by translation hyperovals in the Andre/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2, q(k)) are precisely those that have a scattered F-2 -linear set of pseudoregulus type in PG(2k -1, q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2, q(2))

Exterior splashes and linear sets of rank 3

In \PG(2,q^3), let $\pi$ be a subplane of order $q$ that is exterior to \li. The exterior splash of $\pi$ is defined to be the set of $q^2+q+1$ points on \li that lie on a line of $\pi$. This article investigates properties of an exterior \orsp\ and its exterior splash. We show that the following objects are projectively equivalent: exterior splashes, covers of the circle geometry $CG(3,q)$, Sherk surfaces of size $q^2+q+1$, and \GF(q)-linear sets of rank 3 and size $q^2+q+1$. We compare our construction of exterior splashes with the projection construction of a linear set. We give a geometric construction of the two different families of sublines in an exterior splash, and compare them to the known families of sublines in a scattered linear set of rank 3

Maximum scattered linear sets and MRD-codes

The rank of a scattered -linear set of , rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered -linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered -linear sets of of maximum rank n yield -linear MRD-codes with dimension 2n and minimum distance . We generalize this result and show that scattered -linear sets of of maximum rank rn / 2 yield -linear MRD-codes with dimension rn and minimum distance n - 1

A new family of MRD-codes

We introduce a family of linear sets of PG(1,q^2n) arising from maximum scattered linear sets of pseudoregulus type of PG(3,q^n). For n=3,4 and for certain values of the parameters we show that these linear sets of PG(1,q^2n) are maximum scattered and they yield new MRD-codes with parameters (6,6,q;5) for q>2 and with parameters (8,8,q;7) for q odd