We prove that 45 is the size of the largest caps in AGð5; 3Þ; and such a 45-cap is always obtained from the 56-cap in PGð5; 3Þ by deleting an 11-hyperplane. # 2002 Elsevier Science (USA)

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Let m 0 2 ð3; qÞ be the largest value of k ðkoq 2 þ 1Þ for which there exists a complete k-cap in PGð3; qÞ; q even. In this paper, the known upper bound on m 0 2 ð3; qÞ is improved. We also improve a number of intervals, for k; for which there does not exist a complete k-cap in PGð3; qÞ; q even. r

Abstract:We classify all fdð p 3 þ 1Þ; d; 3;, for a prime number p 0 ! 7, with excess e p 3 . Such a minihyper is a sum of lines and of possibly one projected subgeometry PGð5; pÞ, or a sum of lines and a minihyper which is a projected subgeometry PGð5; pÞ minus one line. When p is a square, also (possibly projected) Baer subgeometries PGð3; p

In [10], it was shown that small t-fold (n − k)-blocking sets in PG(n, q), q = p h , p prime, h ≥ 1, intersect every k-dimensional space in t (mod p) points. We characterize in this article all t-fold (n − k)-blocking sets in PG(n, q), q square, q ≥ 661, t < c p q 1/6 /2, |B| < tq n−k + 2tq n−k−1 √ q, intersecting every k-dimensional space in t (mod √ q) points.