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.

Pseudo-Homogeneous Coordinates for Hughes Planes

Among 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”.

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

The 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

In 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.