Small blocking sets of hermitian designs

AbstractA Hermitian design H(q) consists of the points and Hermitian unitals of PG(2,q2). A committee of H(q) is a blocking set of H(q) of minimum cardinality b(H(q)). It is proved that the committees of H(3) are the lines of PG(2, 9) and, for all odd q, that 2q + 2 ≤b(H(q)) < (1 + 7 ln q)(q2 + 1)q−1

On the isomorphism of certain primitive Q-polynomial not P-polynomial association schemes

In 2011, Penttila and Williford constructed an infinite new family of primitive Q-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space H(3,q2), q even, with respect to a symplectic polar space W(3,q) embedded in it. In a private communication to Penttila and Williford, H. Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of PGL(2,q2), q even, on a non-degenerate conic of PG(2,q2) extended in PG(2,q4). Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs

On symplectic semifield spreads of PG(5,q2), q odd

We prove that there exist exactly three non-equivalent symplectic semifield spreads of PG ( 5 , q2), for q2> 2 .38odd, whose associated semifield has center containing Fq. Equivalently, we classify, up to isotopy, commutative semifields of order q6, for q2> 2 .38odd, with middle nucleus containing q2Fq2and center containing q Fq

A group theoretic characterization of Buekenhout–Metz unitalsin PG(2, q2) containing conics

Let U be a unital in PG(2, q^2), q = p^h and let G be the group of projectivities of PG(2, q2) stabilizing U. In this paper we prove that U is a Buekenhout–Metz unital containing conics and q is odd if, and only if, there exists a point A of U such that the stabilizer of A in G contains an elementary Abelian p-group of order q^2 with no non-identity elations

Covering pairs by q2 + q + 1 sets

AbstractFor given k and s let n(k, s) be the largest cardinality of a set whose pairs can be covered by sk-sets. We determine n(k, q2 + q + 1) if a PG(2, q) exists, k > q(q + 1)2, and the remainder of k divided by (q + 1) is at least √q. Asymptotic results are also given for n(k, s) whenever s is fixed and k → ∞. Our main tool is the theory of fractional matchings of hypergraphs

Nontraditional Positional Games: New methods and boards for playing Tic-Tac-Toe

In this dissertation we explore variations on Tic-Tac-Toe. We consider positional games played using a new type of move called a hop. A hop involves two parts: move and replace. In a hop the positions occupied by both players will change: one will move a piece to a new position and one will gain a piece in play. We play hop-positional games on the traditional Tic-Tac-Toe board, on the finite planes AG(2, q) and PG(2, q) as well as on a new class of boards which we call nested boards. A nested board is created by replacing the points of one board with copies of a second board. We also consider the traditional positional game played on nested boards where players alternately occupy open positions. We prove that the second player has a drawing strategy playing the hop-positional game on AG(2, q) for q ≥ 5 as well as on PG(2, q) for q ≥ 3. Moreover we provide an explicit strategy for the second player involving weight functions. For four classes of nested boards we provide a strategy and thresholds for the second player to force a draw playing a traditional positional game as well as the new hop-positional game. For example we show that the second player has a drawing strategy playing on the nested board [AG(2, q1 ) : PG(2, q2 )] for all q2 ≥ 7. Other bounds are also considered for this and other classes of nested boards

Mixed partitions of PG(3,q2)

AbstractA mixed partition of PG(2n−1,q2) is a partition of the points of PG(2n−1,q2) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q2) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q2n. In this paper, we provide several new examples of such mixed partitions in the case when n=2

Subgeometry partitions from cyclic semifields

New cyclic semifield planes of order qlcm(m,n) are constructed. By varying m and n, while preserving the lcm(m,n), necessarily mutually non-isomorphic semifield planes are obtained. If lcm(m,n)/m = 3, new GL(2,qm) - q3m-planes are constructed. If m is even, new subgeometry partitions in PG(lcm(m, n)-1, q2), by subgeometries isomorphic to either PG(lcm(m,n)/2-1, q2) or PG(lcm(m,n)-1, q) are constructed. If the 2-order of m is strictly larger than the 2-order of n then ‘double’ retraction is possible producing two distinct subgeometry partitions from the same semifield plane. If m is even and lcm(m,n)/m = 3, new subgeometry partitions may be constructed from the GL(2,qm) - q3m-planes