Hughes Plane Research Articles

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR homogeneous.In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive.We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property.Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

On multiple blocking sets and blocking semiovals in finite non-Desarguesian planes

In this paper we first provide an infinite family of minimal (q−1)-fold blocking sets of size q3 in every affine translation plane of order q2.Next, we focus on the regular Hughes plane H(q2) of order q2. It is well known that π=PG(2,q) is embedded as a Baer subplane in H(q2). Let ℓ be a line of H(q2) having q+1 points in common with π and let us denote by H(q2)ℓ the affine plane obtained from H(q2) by deleting ℓ and all the points incident with ℓ. We exhibit a family of minimal (q−1)-fold blocking sets of size q3 in H(q2)ℓ.Furthermore, we consider the polarity of H(q2) which has q3−q2+q2+1 absolute points and we show that its absolute points form a blocking semioval. Finally, we describe a class of Baer subplanes of H(q2) distinct from π and admitting an automorphism group of order q3(q−1).

On PGL(2,q)-invariant unitals embedded in Desarguesian or in Hughes planes

In this paper it is shown that any unital embedded either in PG(2,q2) or in the Hughes plane of order q2, which is invariant under PGL(2,q)×〈σ〉, where σ is the Frobenius involution, is Hermitian or classical Rosati, respectively.

Subplanes of Order $3$ in Hughes Planes

In this study we show the existence of subplanes of order $3$ in Hughes planes of order $q^2$, where $q$ is a prime power and $q \equiv 5 \ (mod \ 6)$. We further show that there exist finite partial linear spaces which cannot embed in any Hughes plane.

Subplanes of projective planes of order 121

We determine orbit representatives of all proper subplanes generated by quadrangles of a Veblen-Wedderburn (VW) plane Π of order 112 and the Hughes plane Σ of order 112 under their full collineation groups. In Π, there are 13 orbits of Baer subplanes all of which are desarguesian and approximately 3000 orbits of Fano subplanes. In Σ, there are 8 orbits of Baer subplanes all of which are desarguesian, 2 orbits of subplanes of order 3 and at most 408,075 distinct Fano subplanes. This work was motivated by the well-known question: “Does there exist a non-desarguesian projective plane of prime order?” The question remains unsettled.

An arc partition of the Hughes plane using a field-theoretic model

An arc partition of the Hughes plane using a field-theoretic model

Unitals in the code of the Hughes plane

AbstractA coding‐theoretic characterization of a unital in the Hughes plane is provided, based on and extending the work of Blokhuis, Brouwer, and Wilbrink in PG(2,q2). It is shown that a Frobenius‐invariant unital is contained in the p‐code of the Hughes plane if and only if that unital is projectively equivalent to the Rosati unital. © 2003 Wiley Periodicals, Inc.

Characterization of 16-dimensional Hughes planes

The well-known finite Hughes planes have compact analoga with 16-dimensional point space. The automorphism group of such a plane is a 36-dimensional Lie group. Theorem: Assume that the compact projective plane $\cal P $ is not isomorphic to the classical Moufang plane over the octonions. Let $\Delta $ be a closed subgroup of $\hbox {Aut} \,\cal P $. If $\dim \Delta \ge 31$ and if $\Delta $ has a normal torus subgroup, then $\cal P $ is a Hughes plane, $\Delta = \hbox {Aut} \,\cal P $, and $\Delta ^{\prime } \cong \hbox {PSL}_3 \Bbb H $.

A New Class of Unitals in the Hughes Plane

A new class of unitals in the Hughes planes is enumerated and classified. The unital obtained by L. A. Rosati is shown to be a member of this class. Their collineation groups are determined and the unitals are sorted by projective equivalence. The dual designs are described and certain members are shown to be self-dual.

Pseudo-Homogeneous Coordinates for Hughes Planes

AbstractAmong the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

A Family of Unitals in the Hughes Plane

We construct a family of unitals in the Hughes plane. We prove that they are not isomorphic with the classical unitals, and in so doing we exhibit a configuration that exists in the latter, but not in the former. This new configurational property of the classical unitals might serve in the future again as an isomorphism test.A particular instance of our construction has appeared in [11]. But it only concerns itself with the case where the matrix involved is the identity, whereas the present article treats the general case of symmetric matrices over a suitable field. Furthermore, [11] does not answer the isomorphism question. It states that (the English translation is ours) “It remains to be seen whether the unitary designs constructed in this note are isomorphic or not with known designs”.

A unital in the hughes plane of order nine

Eight years ago I found four nonisomorphic 2-(28, 4, 1) designs embedded in the Hughes plane of order nine, using a computer. This note gives an algebraic description of one of them.

A connection between Fano configurations and Minkowski planes of odd order

It is shown in this paper that a pair of points contained in a Fano configuration in a projective plane of odd order cannot induce a Minkowski plane. From this result we derive that no pair of points in the Hughes plane of order 9 can induce a Minkowski plane.

Disegni unitari nei piani di Hughes

Let H(q2) be the Hughes plane on the regular nearfield of order q2 whose center is a field of order q. We construct in H(q2) a unital of parameter q.

Subplanes of the Hughes plane of order 25

Etude de la structure interne du plan de Hughes d'ordre 25 et classification de ses sous-plans

The combinatorial structure of the Hughes–Zassenhaus plane of order 25

AbstractThe Hughes–Zassenhaus plane, , of order 25 is the simplest of the exceptional Hughes planes (see, e.g. (1), p. 391).In this paper, an incidence table is constructed analogous to tables for the field plane and the regular Hughes plane, and the following rather surprising properties of the plane emerge:(i) The table has much greater internal symmetry than either of the analogous tables,(ii) any polarity with regard to a conic in the central subplane Δ0 of extends to two pairs of polarities in one pair with 30 and the other with 60 singular points each in addition to the six in Δ0.

The combinatorial structure of the Hughes plane. II

AbstractIn the first part of this paper, tests were described for determining which points of any line in a Galois plane II of order q2 are to be transferred to the conjugate line in order to transmute II into the corresponding Hughes plane Ω. In this part of the paper the tests are refined to provide, in relation to some fixed point in the central subplane δ0 of Ω (i) a simple geometrical condition of transfer for a certain set of ½q(q2−1) points of II and (ii) a simple aglebraic condition for the remaining points of II – δ0. These tests eliminate from the computation (for a given value of q) the necessity of calculating the third coordinates of ½q2 (q2−1) points in order to determine which are not-squares.

Polarities and ovals in the Hughes plane

SummaryIn 1946 Baer (Polarities infinite projective planes, Bull. Am. Math. Soc. 52, 77–93) showed that the absolute points of a polarity in a finite projective plane of odd non-square order always form an oval, that is, in a plane of order n there are exactly n+ 1 absolute points and no three are collinear. It is well known that the absolute points of polarities in planes of odd square order form ovals in some cases.If the oval is a subset of the set of absolute points, then the oval itself determines the polarity, and this makes it appear unlikely that the oval could be a proper subset. Among other results in the paper it is to be proved that in the regular Hughes plane there is a polarity which is determined by an oval which is a relatively small subset of the set of absolute points. Explicitly, if Ω is the Hughes plane of order q2 and A is the central subplane of order q, then every conic in Δ can be extended to an oval in Ω, and this oval determines a polarity in which there are ½(q3–q) additional absolute points.

Generalized Hughes Planes

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:(a) F is finite,(b) the centre and kernel of F coincide,(c) F is of rank 2 over its kernel.(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

The combinatorial structure of the Hughes plane

AbstractThe main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).