More translation planes and semifields from Dembowski–Ostrom polynomials

In this article we use pairs of Dembowski–Ostrom polynomials with special properties (see (P1)–(P3) in the introduction below) to construct translation planes of order q ⁿ which admit cyclic groups of order q ⁿ ?1 having orbits of lengths 1, 1, (q ⁿ ?1)/2, (q ⁿ ?1)/2 on the line at infinity. The same pairs also define semifields of order q ²ⁿ . We discuss the properties of these translation planes and semifields. These constructions extend the related construction in Dempwolff and Müller, Osaka J Math [5].     

A translation plane of order 72 with a very small translation complement

A translation plane of order 7² is constructed whose left nucleus is a cyclic group of order 6. Generally the translation complement of a translation plane is quite large when compared to the order of the left nucleus. The plane discussed in this paper has the distinguished feature that its translation complement modulo the scalar collineations group is very small. In fact it is a Dihedral group of order 12 and is the smallest of all the known examples. The translation complement divides the set of ideal points into 9 orbits of lengths 1,1, 6, 6, 6, 6, 6, 6, and 12.     

Mixed partitions and related designs

We define a mixed partition of Π =  PG(d, q ^r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q ²) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q ²) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q ²ⁿ . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q ^r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r ² n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q ^rn ).     

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1−1 on l ∞

A class of translation planes of order q ^2m+1, where q is an odd prime power and m1, is constructed. If m=1, then this class is contained in the class of order q ³ constructed by Hiramine [5]. These planes of order q ^2m+1 are of dimension 2m+1 over their kernels. If q ^2m+13³, then the linear translation complements of these planes have two orbits of length 2 and q ^2m+1–1 on l and this class contains many planes which are not generalized André planes. If q ^2m+1= 3³, then each plane of this class is isomorphic to the Hering plane of order 27.Dedicated to Professor Tuyosi Oyama on his 60th birthday     

Translation complements of C-planes: (IV)

A new class of finite translation planes called C-planes were constructed by Narayana Rao, Rodabaugh, Wilke and Zemmer from the exceptional near fields of Zassenhaus (Journal of Combinatorial Theory (A) 11, 72–92 (1972)). In this paper we determine the translation complement and its action on the set of ideal points of the C-plane corresponding to the C-system V-1. It is found that the translation complement is of order 5760 and it divides the ideal points into 3 orbits of lengths 2, 24 and 96.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

A translation plane of order 112

A translation plane of order 11² is constructed. Its translation complement is a solvable group of order 1200 and has 9 orbits on the line at infinity. These orbits have lengths 30, 20, 20, 12, 10, 10, 10, 5, 5, respectively.     

Characterization of Translation Planes by Orbit Lengths

The Desarguesian, Hall, and Hering translation planes of order q² are characterized as exactly those translation planes of odd order with spreads in PG (3,q) that admit a linear collineation group with infinite orbits one of length q+1 and i of length (q-q) /i for i=1 or 2.     

A family of translation planes of order q3

In this note, some new class of translation planes of order q³, where q is an odd prime power with q 3,7, are constructed. The translation complement of any plane of this class has three orbits lengths 1, 1 and q³-1 on 1.     

Elementary abelian 2-groups on the line at infinity of translation planes

Let G be a subgroup of the linear translation complement of a translation plane of order q^d with kernel GF(q) and let ¯G be the factor group modulo the scalars. We show that if ¯G is elementary abelian of order 2^a, and if each involution in ¯G has a conjugate class of length greater than a+1 then 2^e divides d, where e=[1/2(a+1)]–1. We show that one of Walker's planes is a counterexample if we drop the condition on lengths of conjugate classes. The Walker plane in question turns out to be of rank 3. This is one of Walker's planes of order 25 and was not previously known to have rank 3.Dedicated to R. Artzy     

Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity

A classification is given of all translation planes of order q ² that admit a collineation group G admitting a two-transitive orbit of q + 1 points on the line at infinity.     

On the integer points on some special hyper-elliptic curves over a finite field

If $\text{l}$_r(p) is the least positive integral value of x for which y² ≡ x(x + 1) ? (x + r ? 1)(modp) has a solution, we conjecture that $\text{l}$_r(p) ≤ r² ? r + 1 with equality for infinitely many primes p. A proof is sketched for r = 5. A further generalization to y² ≡ (x + a₁) ? (x + a_r) is suggested, where the a's are fixed positive integers.     

Spreads,ovoids andS 5

We consider translation planes of orderq ² (whereq andq ² - 1 are coprime to 30) such thatS ₅ acts on the line at infinity. It turns out that the Klein correspondence is in particular useful for the investigation of these planes. Representations of the planes, automorphisms and examples of low order are studied in detail. In view of a problem of Ostrom (Math. Z. 156 (1977), 59–71), series of translation planes are constructed with the following property: the translation complement is nonsolvable and has an order coprime to the characteristic of the plane.     

Embedding (r, 1)-designs in finite projective planes

It is known that any non-trivial (r,1)-design on υ varieties (υ ? (r? 1)² ? 1) is extendible; this fact implies the existence of a projective plane of order r ? 1. In this paper it is shown that any non-trivial (r, 1)-design on (r ? 1)² ? α varieties, where r and α are appropriately bounded, is extendible; hence this fact implies the existence of a projective plane of order r ? 1. We also show that, for υ ? (r ? 1)² ? 2, any non-trivial (r, 1)-design on υ varieties is extendible.     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

A projective plane of order 16

A projective plane of order 16 is constructed. It is a translation plane and appears to be new. The representation of the collineation group on the axis of the plane has a normal subgroup isomorphic to L₃ (2) with factor group isomorphic to S₃. The orbits of this representation have lengths 14 and 3. If two points in the latter orbit are chosen to define a sharply doubly transitive set of permutations, the permutations from the multiplicative loop generate a group isomorphic to A₇. The plane is of Lenz-Barlotti class IVa.1.     

A construction for PBIBD(2)'s

This note gives a new construction for PBIBD(2)'s that generalizes a construction of Hall's for finite projective planes, and that leads to a new PBIBD(2) with parameters (v, b, k, r, λ₁, λ₂) = (36, 60, 10, 0, 2).     

Some New Maximal Sets of Mutually Orthogonal Latin Squares

Two ways of constructing maximal sets of mutually orthogonal Latin squares are presented. The first construction uses maximal partial spreads in PG(3, 4) \ PG(3, 2) with r lines, where r ∈ {6, 7}, to construct transversal-free translation nets of order 16 and degree r + 3 and hence maximal sets of r + 1 mutually orthogonal Latin squares of order 16. Thus sets of t MAXMOLS(16) are obtained for two previously open cases, namely for t = 7 and t = 8. The second one uses the (non)existence of spreads and ovoids of hyperbolic quadrics Q ⁺ (2m + 1, q), and yields infinite classes of q ^{2n ? 1} ? 1 MAXMOLS(q ²ⁿ ), for n ≥ 2 and q a power of two, and for n = 2 and q a power of three.     

Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.