A new hyperoval inPG(2, 32)

A new hyperoval is constructed in the Desarguesian projective planePG(2, 32). It has a full collineation stabiliser of order 3. It is also shown, with the aid of a computer, that the collineation stabiliser of any further new hyperoval is either trivial or has order 2 with the generator fixing no point on the hyperoval.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

A group theoretic characterization of Buekenhout-Metz unitals

It is shown that a unital U embedded in PG(2,q²) is a Buekenhout-Metz unital if and only if U admits a linear collineation group that is a semidirect product of a Sylow p-subgroup of order q³ by a subgroup of order q − 1. This is the full linear collineation group of U except for two equivalence classes of unitals: (i) the classical unitals, and (ii) the Buekenhout-Metz unitals which can be expressed as a union of a partial pencil of conics. The unitals in class (ii) only occur when q is odd, and any two of them are projectively equivalent. © 1996 John Wiley & Sons, Inc.     

Transitive Arcs in Planes of Even Order

When one considers the hyperovals inPG(2,q),qeven,q>2, then the hyperoval inPG(2, 4) and the Lunelli-Sce hyperoval inPG(2, 16) are the only hyperovals stabilized by a transitive projective group [10]. In both cases, this group is an irreducible group fixing no triangle in the plane of the hyperoval, nor in a cubic extension of that plane. Using Hartley's classification of subgroups ofPGL₃(q),qeven [6], allk-arcs inPG(2,q) fixed by a transitive irreducible group, fixing no triangle inPG(2,q) or inPG(2,q³), are determined. This leads to new 18-, 36- and 72-arcs inPG(2,q),q=2^2h. The projective equivalences among the arcs are investigated and each section ends with a detailed study of the collineation groups of these arcs.     

On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

Some geometry of Hermitian matrices of order three over GF(q²) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M₇³of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM₇³ turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q²) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q²) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q²) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q²) new mixed partitions of PG(8,q ) into caps and linear subspaces are given.     

Buekenhout-Tits Unitals

A Buekenhout-Tits unital is defined to be a unital in PG(2, q²) obtained by coning the Tits ovoid using Buekenhout's parabolic method. The full linear collineation group stabilizing this unital is computed, and related design questions are also addressed. While the answers to the design questions are very similar to those obtained for Buekenhout-Metz unitals, the group theoretic results are quite different     

Mixed Partitions of Projective Geometries

Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qⁿ–1)/(q–1) or (qⁿ–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety of PG(8,q) into caps of size q²+q+1 which are Veronese surfaces.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

Planar functions over finite fields

Letp>2 be a prime. A functionf: GF(p)GF(p) is planar if for everyaGF(p) ^*, the functionf(x+a–f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp ², thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.     

Classification of hyperovals inPG(2,32)

In this paper, we describe an exhaustive computer search that demonstrates that there are precisely 6 isomorphism classes of hyperovals inPG(2,32). The six classes had previously been discovered, and it was known that any further hyperovals would have stabiliser groups of orders 1 or 2. As the techniques for finding hyperovals involved a mixture of group theory and computer search, an exhaustive search was regarded as the only feasible way to eliminate these final cases with small group.     

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.     

On PSL(2,q) as a totally irregular collineation group

Non-abelian simple totally irregular collineation groups containing an involutorial perspectivity have been classified by the authors in a recent paper. They are PSL(2,q), PSL(3,q), PSU(3,q), Sz(q), the alternating group on 7 letters, and the second Janko sporadic simple group. In this article, we study PSL(2,q),q congruent to 1 modulo 4, as a collineation group containing an involutory homology.C. Y. Ho was partially supported by a NSA grant.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

Computer search in projective planes for the sizes of complete arcs

The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group PΓ L (3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809, q ≠ 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167.     

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q ⁴) (and show infinite versions of this embedding). Dan Hughes 80th Birthday.     

Orthogonal packings inPG(5, 2)

Aspread inPG(n, q) is a set of lines which partitions the point set. A packing inPG(n, q) (n odd) is a partition of the lines into spreads. Two packings ofPG(n, q) are calledorthogonal if and only if any two spreads, one from each packing, have at most one line in common. Recently, R. D. Baker has shown the existence of a pair of orthogonal packings inPG(5, 2). In this paper we enumerate all packings inPG(5, 2) having both an automorphism of order 31 and the Frobenius automorphism. We find all pairs of orthogonal packings of the above type and display a set of six mutually orthogonal packings. Previously the largest set of orthogonal packings known inPG(5, 2) was two.     

Geometry of the Mathieu groups and Golay codes

A brief review is given of the linear fractional subgroups of the Mathieu groups. The main part of the paper then deals with the projective interpretation of the Golay codes; these codes are shown to describe Coxeter’s configuration inPG(5,3) and Todd’s configuration inPG(11,2) when interpreted projectively. We obtain two twelve-dimensional representations ofM ₂₄. One is obtained as the collineation group that permutes the twelve special points inPG(11,2); the other arises by interpreting geometrically the automorphism group of the binary Golay code. Both representations are reducible to eleven-dimensional representations ofM ₂₄.     

A structure theory for two-dimensional translation planes of order q 2 that admit collineation groups of order q 2

This paper is devoted to the study of translation planes of order q ² and kernel GF(q) that admit a collineation group of order q ² in the linear translation complement. We give a representation of this group by a suitable set of matrices depending on some functions over GF(q). Using this representation we obtain several results concerning the existence and the collineation group of the plane.     

Near-MDS codes arising from algebraic curves

Every elliptic quartic Γ₄ of PG(3,q) with nGF(q)-rational points provides a near-MDS code C of length n and dimension 4 such that the collineation group of Γ₄ is isomorphic to the automorphism group of C. In this paper we assume that GF(q) has characteristic p>3. We classify the linear collineation groups of PG(3,q) which can preserve an elliptic quartic of PG(3,q). Also, we prove for q?113 that if the j-invariant of Γ₄ does not disappear, then C cannot be extended in a natural way by adding a point of PG(3,q) to Γ₄.     

Transitive Partial Parallelisms of Deficiency One

A new construction of parallelisms, determined by Johnson, is valid for both the finite and infinite cases and gives a variety of partial parallelisms of deficiency one that admit a transitive group. Since there are extensions to parallelisms, one obtains parallelisms admitting a collineation group fixing one spread and transitive on the remaining spreads. The construction permits a counting of the isomorphism classes of the parallelisms. In this article, we enumerate the isomorphism classes of the parallelisms and show that there are at least 1  +  [(q −  3) / 2 r ] mutually non-isomorphic parallelisms in PG(3,q  =  p^r), for p odd. Furthermore, we provide a group-theoretic characterization of the constructed parallelisms.