Satoshi Yoshiara

Ambient Spaces of Dimensional Dual Arcs

A d-dimensional dual arc in PG(n, q) is a higher dimensional analogue of a dual arc in a projective plane. For every prime power q other than 2, the existence of a d-dimensional dual arc (d ≥ 2) in PG(n, q) of a certain size implies n ≤ d(d + 3)/2 (Theorem 1). This is best possible, because of the recent construction of d-dimensional dual arcs in PG(d(d + 3)/2, q) of size ∑d−1i=0qi, using the Veronesean, observed first by Thas and van Maldeghem (Proposition 7). Another construction using caps is given as well (Proposition 10).

A Family of d -dimensional Dual Hyperovals in P G(2 d+ 1, 2)

With the affine part of an oval we associate a family of d -subspaces ofPG (2 d+ 1, 2) which can be thought of as a higher dimensional analogue of a hyperoval. The isomorphisms among such families together with their automorphisms are determined when they come from translation ovals.

Some Projective Modules Determined by Sporadic Geometries

Abstract We determine which known sporadic geometries have projective Lefschetz modules. An elementary lemma on local collapsibility greatly simplifies the task of verifying projectivity. The modules are analyzed when possible in terms of projective covers of individual irreducibles.

Embeddings of flag-transitive classical locally polar geometries of rank 3

The group-admissible embeddings of flag-transitive classical locally polar geometries of rank 3 are determined, as well as those of truncations of the related dual polar spaces.

Flag-Transitive Buekenhout Geometries

Publisher Summary This chapter discusses the concept of flag-transitive Buekenhout geometries. More than one half of sporadic simple groups are known to act flag-transitively on finite geometries belonging to Buekenhout diagrams obtained from Coxeter diagrams replacing some of strokes for projective planes with strokes for circular spaces or for dual circular spaces. The chapter recalls that a circular space is a finite linear space with lines of size 2—namely, the system of vertices and edges of a complete graph. Classification theorems exist for some classes of geometries; the chapter surveys some of those theorems, choosing certain diagrams for which a classification is known or at least substantial progresses have been done in that direction. The chapter also discusses some background on Lie diagrams, the Neumaier geometry, spherical and nonspherical diagram, and affine constructions.

A geometric characterization of the groups Suz and HS

Buekenhout and Hubaut [BH] studied finite connected c.C,-geometries (more generally, c.C,,-geometries) r whose point-residues are classical quadrangles, admitting a flag-transitive group G whose point stabilizer induces an automorphism group containing the corresponding classical simple group. In their Lemma 7.3, however, they overlooked the group M,, as a transitive extension of M,, acting on the 10 points of a line in a U,(3)quadrangle H,(3’). There is, in fact, an interesting c.C,-geometry associated with the sporadic Suzuki group Suz with such point residues (see [BF], [Bu]). This is a 3-local geometry; it was described in [St]; [Ro, 7.2, pp,326-3271. The set 9 is a certain conjugacy class of subgroups of order 3 in SUZ, and the sets 9 and 9 consist of pairs and maximal sets of mutually commuting subgroups in 9”, respectively. Recently, A. Del Fra, D. Ghinelli, T. Meixner, and A. Pasini [FGMP] studied c.C,-geometries whose point-residues are classical thick quadrangles, admitting a flag-transitive group G. They succeeded in classifying these geometries, except in the case where point-residues are again isomorphic to the U,(3)-quadrangle H,(3’).

A Construction of Differentially 4-Uniform Functions from Commutative Semifields of Characteristic 2

We construct differentially 4-uniform functions over GF(2n) through Alberts finite commutative semifields, if nis even. The key observation there is that for some kwith 0 ≤ k≤ ni¾? 1, the function

Dimensional dual hyperovals associated with Steiner systems

f_{k}(x):=(x^{2^{k+1}}+x)/(x^{2}+x)

The radical subgroups of the Fischer simple groups

is a two to one map on a certain subset D k (n) of GF(2n). We conjecture that f k is two to one on D k (n) if and only if (n,k) belongs to a certain list. For (n,k) in this list, f k is proved to be two to one. We also prove that if f 2 is two to one on D 2 (n) then (n,2) belongs to the list.

A Locally Polar Geometry Associated With the Group HS

In Adv. Geom. 3 (2003) 245, a class of d-dimensional dual hyperovals is constructed starting from a subset X of PG(d, 2) with certain properties. In this paper, a criterion for X to provide a d-dimensional dual hyperoval is given in terms of some functions. Based on this, we describe such subsets, and show that there are exactly two isomorphism classes of d-dimensional dual hyperovals arising from those subsets and that a similar statement holds for those of the associated Steiner systems S(3, 4, 2d+1).