Transitive and Co-Transitive Caps

A cap in PG(r,q) is a set of points, no three of which are collinear. A cap is said to be transitive if its automorphism group in PGammaL(r+1,q) acts transtively on the cap, and co-transitive if the automorphism group acts transtively on the cap's complement in PG(r,q). Transitive, co-transitive caps are characterized as being one of: an elliptic quadric in PG(3,q); a Suzuki-Tits ovoid in PG(3,q); a hyperoval in PG(2,4); a cap of size 11 in PG(4,3); the complement of a hyperplane in PG(r,2); or a union of Singer orbits in PG(r,q) whose automorphism group comes from a subgroup of GammaL(1,q^{r+1}).Comment: To appear in The Bulletin of the Belgian Mathematical Society - Simon Stevi

On curves covered by the Hermitian curve

For each proper divisor d of (r^2-r+1), r being a power of a prime, maximal curves over a finite field with r^2 elements covered by the Hermitian curve of genus 1/2((r^2-r+1)/d-1) are constructed.Comment: 18 pages, Latex2

On line covers of finite projective and polar spaces

An $m$-$cover$ of lines of a finite projective space ${\rm PG}(r,q)$ (of a finite polar space $\cal P$) is a set of lines $\cal L$ of ${\rm PG}(r,q)$ (of $\cal P$) such that every point of ${\rm PG}(r,q)$ (of $\cal P$) contains $m$ lines of $\cal L$, for some $m$. Embed ${\rm PG}(r,q)$ in ${\rm PG}(r,q^2)$. Let $\bar{\cal L}$ denote the set of points of ${\rm PG}(r,q^2)$ lying on the extended lines of $\cal L$. An $m$-cover $\cal L$ of ${\rm PG}(r,q)$ is an $(r-2)$-dual $m$-cover if there are two possibilities for the number of lines of $\cal L$ contained in an $(r-2)$-space of ${\rm PG}(r,q)$. Basing on this notion, we characterize $m$-covers $\cal L$ of ${\rm PG}(r,q)$ such that $\bar{\cal L}$ is a two-character set of ${\rm PG}(r,q^2)$. In particular, we show that if $\cal L$ is invariant under a Singer cyclic group of ${\rm PG}(r,q)$ then it is an $(r-2)$-dual $m$-cover. Assuming that the lines of $\cal L$ are lines of a symplectic polar space ${\cal W}(r,q)$ (of an orthogonal polar space ${\cal Q}(r,q)$ of parabolic type), similarly to the projective case we introduce the notion of an $(r-2)$-dual $m$-cover of symplectic type (of parabolic type). We prove that an $m$-cover $\cal L$ of ${\cal W}(r,q)$ (of ${\cal Q}(r,q)$) has this dual property if and only if $\bar{\cal L}$ is a tight set of an Hermitian variety ${\cal H}(r,q^2)$ or of ${\cal W}(r,q^2)$ (of ${\cal H}(r,q^2)$ or of ${\cal Q}(r,q^2)$). We also provide some interesting examples of $(4n-3)$-dual $m$-covers of symplectic type of ${\cal W}(4n-1,q)$.Comment: 20 page

ORBIT CODES FROM FORMS ON VECTOR SPACES OVER A FINITE FIELD

In this paper we construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an n-dimensional vector space over the finite field Fq. All these codes admit the general linear group GL(n, q) as a transitive automorphism group

Fuzzy Authentication using Rank Distance

Fuzzy authentication allows authentication based on the fuzzy matching of two objects, for example based on the similarity of two strings in the Hamming metric, or on the similiarity of two sets in the set difference metric. Aim of this paper is to show other models and algorithms of secure fuzzy authentication, which can be performed using the rank metric. A few schemes are presented which can then be applied in different scenarios and applications.Comment: to appear in Cryptography and Physical Layer Security, Lecture Notes in Electrical Engineering, Springe

Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$ and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order $(s^2,s)$ for $s \le 11$. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle $H(3,q^2)$.Comment: slight revisions made following referee's reports, and included raw dat

Families of twisted tensor product codes

Using geometric properties of the variety \cV_{r,t}, the image under the Grassmannian map of a Desarguesian $(t-1)$-spread of \PG(rt-1,q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We exactly determine the parameters of these codes and characterise the words of minimum weight.Comment: Keywords: Segre Product, Veronesean, Grassmannian, Desarguesian spread, Subgeometry, Twisted Product, Constacyclic error correcting code, Minimum weigh