Maximum Scattered Linear Sets and Complete Caps in Galois Spaces

Small complete caps in Galois affine spaces

Some new families of caps in Galois affine spaces AG(N , q) of dimension N ≡ 0 (mod 4) and odd order q are constructed. Such caps are proven to be complete by using some new ideas depending on the concept of a regular point with respect to a complete plane arc. As a corollary, an improvement on the currently known upper bounds on the size of the smallest complete caps in AG(N , q) is obtained.

The geometry of covering codes: small complete caps and saturating sets in Galois spaces

Large caps in projective Galois spaces

If the underlying field is F2, the answer is easy: AG(n,2) is itself a cap of 2n points and it forms up to projective equivalence the unique largest cap in PG(n,2). Assume therefore we work in PG(n,q) or AG(n,q) for q > 2. The canonical models for caps are quadrics of Witt index 1. They yield (q+1)-caps in AG(2,q) (and in PG(2,q)) and obviously these ovals are maximal for odd q. In odd characte...

Scattered Linear Sets and Pseudoreguli

In this paper, we show that one can associate a pseudoregulus with every scattered linear set of rank 3n in PG(2n − 1, q3). We construct a scattered linear set having a given pseudoregulus as associated pseudoregulus and prove that there are q − 1 different scattered linear sets that have the same associated pseudoregulus. Finally, we give a characterisation of reguli and pseudoreguli in PG(3, ...

1-saturating Sets, Caps and round Sets in Binary Spaces

We show that, for a positive integer r, every minimal 1-saturating set in PG(r − 1, 2) of size at least 11 36 2r + 3 either is a complete cap or can be obtained from a complete cap S by fixing some s ∈ S and replacing every point s ∈ S \ {s} by the third point on the line through s and s. Stated algebraically: if G is an elementary abelian 2-group and a set A ⊆ G \ {0} with |A| > 11 36 |G| + 3 ...