Fernanda Pambianco

On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists

Abstractthe aim of this paper is to collect all results on the spectrum of values k that occur as the cardinality of a complete k- cap in a finite projective space. 1

Small Complete Caps in Spaces of Even Characteristic

In 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of orderformula]. Generalizing the construction of Segre, complete (qn+3(qn?1+?+q)+2)-caps inPG(2n, q),qeven,q?4, and complete (3(qn+?+q)+2)-caps inPG(2n+1, q),qeven,q?4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of orderformula].

On sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 = 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.

On saturating sets in projective spaces

Minimal saturating sets in projective spaces PG(n, q) are considered. Estimates and exact values of some extremal parameters are given. In particular the greatest cardinality of a minimal 1-saturating set has been determined. A concept of saturating density is introduced. It allows to obtain new lower bounds for the smallest minimal saturating sets. A number of exhaustive results for small q are obtained. Many new small 1-saturating sets in PG(2, q), q≤ 587, are constructed by computer.

Complete arcs in PG(2,25): The spectrum of the sizes and the classification of the smallest complete arcs

In this paper it has been verified, by an exhaustive computer search, that in PG(2,25) the smallest size of a complete arc is 12 and that complete 19-arcs and 20-arcs do not exist. Therefore, the spectrum of the sizes of the complete arcs in PG(2,25)is completely determined. The classification of the smallest complete arcs is also given: the number of non-equivalent complete 12-arcs is 606 and for each of them the automorphism group has been found and some geometrical properties have been studied. The exhaustive search has been feasible because projective equivalence properties have been exploited to prune the search tree and to avoid generating too many isomorphic copies of the same arc.

On Complete Arcs Arising from Plane Curves

AbstractWe point out an interplay between

On some 10-arcs for deriving the minimum order for complete arcs in small projective planes

F_q

The smallest size of a complete cap in PG(3,7)

-Frobenius non-classical plane curves and complete