We construct a complete $11$--span of $\mathcal{H}(4,4)$ admitting the group $\mathrm{PSL}_2(11)$. This span turns out to be associated with the unique Hadamard design $\mathcal{H}_{11}$ and the so-called Petersen design.

This paper is concerned with the teaching of Discrete Mathematics to university undergraduate students. Two to three decades ago this course became a requirement for math and computer science students in most universities world wide. Today this course is taken by students in many other disciplines as well. The paper begins with a discussion of a few topics that we feel should be included in the syllabus for any course in Discrete Mathematics, independent of the audience. We then discuss several potential models for teaching the course, depending upon the interests and mathematical background of the audience. We also investigate various educational links with other components of the curriculum, consider pedagogical issues associated with the teaching of discrete mathematics, and discuss some logistical and psychological difficulties that must be overcome. A special emphasis is placed on the role of textbooks.

Let L be a general linear complex in PG (3, q ) for any prime power q . We show that when GF ( q ) is extended to GF ( q 2 ), the extended lines of L cover a non-singular Hermitian surface H ≅ H (3, q 2 ) of PG (3, q 2 ). We prove that if S is any symplectic spread PG (3, q ), then the extended lines of this spread form a complete ( q 2 + 1)-span of H . Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete ( q 3 + 1)-spans of the Hermitian variety H (5, q 2 ).

We present a proof for a conjecture of De Caen and Van Dam (2001, Europ. J. Combinatorics,22, 297–301) concerning the existence of a four-class association scheme on the set of all unordered pairs of points of the projective line PG(1, q2), where q= 2m.

A hyperbolic fibration is a set of q - 1 hyperbolic quadrics and two lines which together partition the points of PG(3, q). The classical example of a hyperbolic fibration comes from a pencil of quadrics; however, several other families are now known. In this paper we begin the development of a general framework to study hyperbolic fibrations for odd prime powers q.One byproduct of hyperbolic fibrations is the 2(q-1) (not necessarily inequivalent) spreads of PG(3,q) they spawn via the selection of one ruling family of lines for each of the hyperbolic quadrics. We show how the hyperbolic fibration context can be used to unify the study of these spreads, especially those associated with j-planes. The question of whether a spread spawned from such a fibration could contain any reguli other than the ones it inherits from the fibration plays a significant role in the determination of its automorphism group, as well as being an interesting geometric question in its own right. This information is then used to address the problem of sorting out projective equivalences among the spreads spawned from a given hyperbolic fibration. Plucker coordinates are an important tool in most of these investigations.

A complete characterization of reguli that are contained in Singer line orbits is given. The characterization is field theoretic and depends upon modeling PG(3,q) by the finite field GF(q4), viewed as a four-dimensional vector space over GF(q). As applications of this characterization one is able to construct various balanced incomplete block designs and group divisible designs, the most interesting one having the parameters of a (q+1)-fold cover of an inversive plane. A robust method for constructing large families of mutually inequivalent unembeddable translation nets of order q2 and deficiency q is also given.

We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q(2)), for q greater than or equal to 4 a power of 2, thereby proving a conjecture of D. de Caen and E. van Darn (Fissioned triangular schemes via the cross-ratio, European J. Combin. 22 (2001), 297-301). This is a fusion of certain relations in the fission scheme FT(q(2) + 1) obtained from the triangular association scheme. Combining three relations in the above four-class association scheme yields a strongly regular graph, which we show is isomorphic to one constructed by Brouwer and Wilbrink using hyperbolic solid sections of the parabolic quadric in PG(4, q). (C) 2001 Academic Press.

For any odd prime power q, all (q2−q+1)th roots of unity clearly lie in the extension field Fq6 of the Galois field Fq of q elements. It is easily shown that none of these roots of unity have trace −2, and the only such roots of trace −3 must be primitive cube roots of unity which do not belong to Fq. Here the trace is taken from Fq6 to Fq. Computer based searching verified that indeed −2 and possibly −3 were the only values omitted from the traces of these roots of unity for all odd q⩽200. In this paper we show that this fact holds for all odd prime powers q. As an application, all odd order three-dimensional flag-transitive affine planes admitting a cyclic transitive action on the line at infinity are enumerated.

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)).

The classification of perfect Baer subplane partitions of { PG}(2, q^2) is equivalent to the classification of 3-dimensional flag-transitive planes whose translation complements contain a linear cyclic group acting regularly on the line at infinity. Since all known flag-transitive planes admit a translation complement containing a linear cyclic subgroup which either acts regularly on the points of the line at infinity or has two orbits of equal size on these points, such a classification would be a significant step towards the classification of all 3-dimensional flag-transitive planes. Using linearized polynomials, a parametric enumeration of all perfect Baer subplane partitions for odd q is described. Moreover, a cyclotomic conjecture is given, verified by computer for odd prime powers q<200 , whose truth would imply that all perfect Baer subplane partitions arise from a construction of Kantor and hence the corresponding flag-transitive planes are all known.

The aim is to find the maximum size of a set of mutually ske lines on a nonsingular Hermitian surface in PG(3, q) for various values of q. For q = 9 such extremal sets are intricate combinatorial structures intimately connected ith hemisystems, subreguli, and commuting null polarities. It turns out they are also closely related to the classical quartic surface of Kummer. Some bounds and examples are also given in the general case.

A hyperbolic fibration is set of q - 1 hyperbolic quadrics and two lines which together partition the points of PG(3, q). The classical example of a hyperbolic fibration comes from a pencil of quadrics; however, several other families are known. In this paper we construct a new family of hyperbolic fibrations for odd prime powers q.As an application of hyperbolic fibrations, we note that they can be used to construct 2(q-1) (not necessarily inequivalent) spreads of PG(3, q) by choosing one ruling family from each of the hyperbolic quadrics in the fibration. For our new fibration we discuss some properties of the spreads obtained in the above manner. (C) 1999 Academic Press.

This paper is concerned with constructing caps embedded in line Grassmannians. In particular, we construct a cap of size q(3) + 2q(2) + 1 embedded in the Klein quadric of PG(5, q) for even q, and show that any cap maximally embedded in the Klein quadric which is larger than this one must have size equal to the theoretical upper bound, namely q(3) + 2q(2) + q + 2. It is not known if caps achieving this upper bound exist for even q > 2.

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

Determining the clique number of the Paley graph of order q, q ≡ 1 (mod 4) a prime power, is a difficult problem. However, the work of Blokhuis implies that in the Paley graph of order q2, where q is any odd prime power, the clique number is in fact q. In this paper we construct maximal cliques of size 12(q + 1) or 12(q + 3), accordingly as q ≡ 1 (mod 4) or q ≡ 3 (mod 4), in the Paley graph of order q2. 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,q) for odd prime powers q.

It is shown that a unital U embedded in PG(2, q(2)) 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 q(3) 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. (C) 1996 John Wiley & Sons, Inc.

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.

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 zero or two reguli of P. This generalizes the notion of a chain of reguli first introduced by Bruen. If U denotes the lines of S contained in the reguli of P and V is a partial spread of PG(3, q) covering the same points as U, then (S\U) ∪V is a spread of PG(3, q) yielding a (potentially new) translation plane which is 2-dimensional over its kernel. In this paper we construct replaceable nests of size q − 1 for each odd prime power, q, thereby generating an (apparently new) infinite family of translation planes of order q2. The geometric properties as well as collineation groups of the corresponding spreads are studied in some detail. In addition a very interesting relationship between these nests and the known chains of reguli is discovered. This relationship could lead to the construction of more chains and hence more translation planes.

One way to obtain a (hopefully) new nondesarguesian translation plane is by constructing a new spread which is not subregular. Chains of reguli in a regular spread of PG(3, q) were first introduced by Bruen as a device to obtain in a fairly large replaceable partial spread, and hence such a nonsubregular spread. In this paper, we describe various operations on chains enabling us to construct an extremely rich class of replaceable partial spreads, at least for small values of q. We also give a new chain for q = 17.