Geertrui Van de Voorde

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, qh) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, qh).

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73-92], the codewords of small weight in the dual code of the code of points and lines of Q(4,q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q^+(5,q) and H(5,q^2), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q^+(5,q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4,q), q even. To prove this, we show that a blocking set of Q(4,q), q even, of size q^2+1+r, where 0

Extending pseudo-arcs in odd characteristic

Abstract A pseudo-arc in PG ( 3 n − 1 , q ) is a set of ( n − 1 ) -spaces such that any three of them span the whole space. A pseudo-arc of size q n + 1 is a pseudo-oval . If a pseudo-oval O is obtained by applying field reduction to a conic in PG ( 2 , q n ) , then O is called a pseudo-conic . We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens–Szekeres GQ is obtained from a q -arc in PG ( 2 , q ) and we extend this construction to that of a GQ of order ( q n − 1 , q n + 1 ) from a pseudo-arc of PG ( 3 n − 1 , q ) of size q n . The main theorem of this paper shows that if K is a pseudo-arc in PG ( 3 n − 1 , q ) , q odd, of size larger than the size of the second largest complete arc in PG ( 2 , q n ) , where for one element K i of K , the partial spread S = { K 1 , … , K i − 1 , K i + 1 , … , K s } / K i extends to a Desarguesian spread of PG ( 2 n − 1 , q ) , then K is contained in a pseudo-conic. The main result of Casse et al. (1985) [5] also follows from this theorem.

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e., point sets of size

On the maximality of a set of mutually orthogonal Sudoku Latin Squares

q+t

Unitals with many Baer secants through a fixed point

q+t in

On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

\mathrm {PG}(2,q)