R. D. Baker

On Buekenhout-Metz unitals of odd order

Abstract The odd order Buekenhout-Metz unitals are enumerated and classified. Their inherited collineation groups are computed, they are shown to be self-dual as designs, and related designs are constructed.

An elliptic semiplane

Abstract An elliptic semiplane (symmetric group divisible design with λ 1 = 1 and λ 2 = 0) is constructed. This elliptic semiplane cannot be realized as a projective plane minus a Baer subset, and is the first elliptic semiplane constructed which has this property.

Mixed partitions of PG(5, q )

Abstract We prove that the projective space PG(5, q ) can be partitioned into two planes and q 3 −1 caps all of which are quadric Veroneseans. This partition is obtained by taking the orbits of a lifted Singer cycle of PG(2, q ). The possibility of getting larger caps by gluing some of these orbits together is also addressed.

A New Class of Translation Planes

We define a nest of reguli to be a collection P of reguli in a regular spread S of PG(3, q) such that every line of S is contained in exactly 0 or 2 reguli of P. Let U denote the lines of S contained in the reguli of some nest. If V is a partial spread of PG[3,q) covering the same points as U but having no lines in common with U , then V will be called a replacement set for U. Clearly, (S—U) U V is a spread of PG(3, q) , yielding a (potentially new) translation plane of order q 2 which is 2–dimensional over its kernel. Nests of size (q+3)/2 were first studied (under another name) by Bruen and later by many others. Whether such (q+3)/2- nests exist for q 13 and whether such nests are necessarily reversible are still open questions. In this paper we consider nests of size q. We exhibit an infinite family of ?-nests, one for each odd prime q , and show that each nest is reversible. The translation planes so obtained appear to be new, at least for q ≥ 11.

Quasigroups and tactical systems

A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schröders Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Steins Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Steins Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.

Two-dimensional flag-transitive planes revisited

This paper shows that the odd order two-dimensional flag-transitive planes constructed by Kantor-Suetake constitute the same family of planes as those constructed by Baker-Ebert. Moreover, for orders satisfying a modest number theoretical assumption this family consists of all possible such planes of that order. In particular, it is shown that the number of isomorphism classes of (non-Desarguesian) two-dimensional flag-transitive affine planes of order q2 is precisely (q−1)/2 when q is an odd prime and precisely (q−1)/2e when q=pe is an odd prime power with exponent e that is a power of 2. An enumeration is given in other cases that uses the Möbius inversion formula.

Maximal cliques in the Paley graph of square order

Copyright (c) 1996 Elsevier Science B.V. All rights reserved. Determining the clique number of the Paley graph of order q, q≡1 (mod 4r a prime power, is a difficult problem. However, the work of Blokhuis implies that in the Paley graph of order q 2 , where q is any odd prime power, the clique number is in fact q. In this paper we construct maximal cliques of size (1r/(2r(q+1r or (1r/(2r(q+3r, accordingly as q≡1 (mod 4r or q≡3 (mod 4r, in the Paley graph of order q 2 . It is believed that these are the largest maximal cliques which are not maximum. We also briefly discuss maximal cliques in some graphs naturally associated with the interior and exterior points of a conic in PG(2,qr for odd prime powers q.

Construction of two-dimensional flag-transitive planes

An affine plane is called flag-transitive if it admits a collineation group which acts transitively on the incident point-line pairs. It has been shown that finite flag-transitive planes are necessarily translation planes, and much work has been devoted to this class of translation planes in recent years. All flag-transitive groups of finite affine planes have been determined, and an infinite family of non-Desarguesian flag-transitive planes has been found. In this paper a method is given for constructing all two-dimensional flag-transitive planes of odd order, subsuming the infinite family mentioned above.

Resolvable bibd and sols

The ideas of a base factorization and a resolvable grid system are introduced, and a construction of a resolvable balanced incomplete block design (BIBD) from these structures is given. Resolvable grid systems can be constructed from mutually orthogonal self-orthogonal latin squares (SOLS) with symmetric mate. Together these results prove, as a special case, that: if k-1 is an odd prime power and there exist12(k-2) mutually orthogonal SOLS of order n, with symmetric mate, then there exists a resolvable BIBD with block size k on v = kn points of index @l, where @l = k-1 ifk = 0 (mod 4) and @l = 2(k-1) if k = 2 (mod 4). The technique is illustrated for k = 4, @l = 3 and k = 6, @l = 10, in which cases v = 0 (mod k) is shown to be a necessary and sufficient condition (NASC) for the existence of a resolvable BIBD on v points. The pair (k, @l) = (6,10) thus becomes only the fifth pair for which NASC are known, the other pairs being (3,1), (4,1), (3,2), and (4,3).

Hyperbolic Fibrations and q-Clans

We show there is a bijection between regular hyperbolic fibrations with constant back half and normalized q-clans. Thus there is also a bijection with flocks of a quadratic cone, once a conic of the flock has been specified. This yields a plethora of two-dimensional translation planes of even and odd order which arise from spreads admitting a regular elliptic cover.