Finite Translation Planes Research Articles

Scattered linear sets in a finite projective line and translation planes

In [15], Lunardon and Polverino construct a translation plane starting from a scattered linear set of pseudoregulus type in PG(1,qt). In this paper a similar construction of a translation plane Af obtained from any scattered linearized polynomial f(x) in Fqt[x] is described and investigated. A class of quasifields giving rise to such planes is defined. Denote by Uf the Fq-subspace of Fqt2 associated with f(x). If f(x) and f′(x) are scattered, then Af and Af′ are isomorphic if and only if Uf and Uf′ belong to the same orbit under the action of ΓL(2,qt). This gives rise to the same number of distinct translation planes as the number of inequivalent scattered linearized polynomials. In particular, for any scattered linear set L of maximum rank in PG(1,qt) there are cΓ(L) pairwise non-isomorphic translation planes, where cΓ(L) denotes the ΓL-class of L, as defined in [5] by Csajbók, Marino and Polverino. A result by Jha and Johnson [8] allows to describe the automorphism groups of the planes obtained from the linear sets not of pseudoregulus type defined in [15].

Problems on structure of finite quasifields and projective translation planes

It is well-known that the constructions and classification of non-Desarguesian projective planes are closely connected with ones for quasifields. We consider the problems on structure of finite quasifields and semifields: automorphisms and autotopisms, maximal subfields and their orders, the spectrum of orders of non-zero elements and hypotheses about generated subsets of the multiplicative loop.

Subplanes of a translation plane

In this article we determine conditions for a finite translation plane π in order to contain subplanes and derivation sets. We study the effects of transposition on affine subplanes of π introducing a geometric procedure to construct, starting from a Baer subplane π0 of π, one π0′ lying in the transpose planeπt of π. We prove that, in general, π0′ is isomorphic neither to π0 nor to the transpose π0t of π0. Also, in the semifield case, we prove that some Knuth derivatives of the recently discovered Budaghyan–Helleseth semifield planes are derivable and hence derive to a strict translation plane. Our analysis further shows that any plane coordinatized by a commutative Albert semifield may be embedded in a Budaghyan–Helleseth plane.

A Generalization of Some Huang–Johnson Semifields

In [H. Huang, N.L. Johnson: Semifield planes of order $8^2$, Discrete Math., 80 (1990)], the authors exhibited seven sporadic semifields of order $2^6$, with left nucleus ${\mathbb F}_{2^3}$ and center ${\mathbb F}_2$. Following the notation of that paper, these examples are referred as the Huang–Johnson semifields of type $II$, $III$, $IV$, $V$, $VI$, $VII$ and $VIII$. In [N. L. Johnson, V. Jha, M. Biliotti: Handbook of Finite Translation Planes, Pure and Applied Mathematics, Taylor Books, 2007], the question whether these semifields are contained in larger families, rather then sporadic, is posed. In this paper, we first prove that the Huang–Johnson semifield of type $VI$ is isotopic to a cyclic semifield, whereas those of types $VII$ and $VIII$ belong to infinite families recently constructed in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti: Semifields of order $q^6$ with left nucleus ${\mathbb F}_{q^3}$ and center ${\mathbb F}_q$, Finite Fields Appl., 14 (2008)] and [G.L. Ebert, G. Marino, O. Polverino, R. Trombetti: Infinite families of new semifields, Combinatorica, 6 (2009)]. Then, Huang–Johnson semifields of type $II$ and $III$ are extended to new infinite families of semifields of order $q^6$, existing for every prime power $q$.

On the planes of Suetake

The Suetake planes and the simple full non-square Andre planes are shown to be exactly those finite translation planes that are coupled with a semifield plane. Also, the Suetake planes are never generalized Andre planes.

An upper bound for the minimum weight of the dual codes of desarguesian planes

We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529–538] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p m where p is a prime, and m ≥ 1 . This gives words of weight 2 p m + 1 − p m − 1 p − 1 in the dual of the p -ary code of the desarguesian plane of order p m , and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [Gábor Korchmáros, Francesco Mazzocca, On ( q + t ) -arcs of type ( 0 , 2 , t ) in a desarguesian plane of order q , Math. Proc. Cambridge Philos. Soc. 108 (1990) 445–459].

Groups generated by affine perspectivities

Abstract Let G be a collineation group of a finite translation plane such that G is generated by perspectivities fixing a point 0. Let M be a non-abelian minimal normal subgroup of G. We show that M is uniquely determined, that the translation plane has characteristic 2, and that M ≅ L 2(q), q even, or M ≅ Sz(q).

SOME CHARACTERIZATIONS OF DESARGUESIAN TRANSLATION PLANES

In this note we consider finite translation planes with large translation complements. In particular, we characterize finite affine Desarguesian translation planes in two ways, according to the existence of subgroups in the translation complement that are divisible by relatively large integers, together with modest additional restrictions.

Simple collineation groups with perspectivities of finite translation planes

Simple collineation groups with perspectivities of finite translation planes

Functions on Linear Spaces Associated with Finite Projective Planes

Using spread sets that define finite translation planes, we construct functions that map a finite linear space into itself. Properties of such functions, which are of interest from the standpoint of cryptography, are examined. We look into the relationship between these functions and corresponding translation planes.

The translation planes admitting a nonsolvable doubly transitive line-sized orbit

The finite translation planes of ordern not in {34,36,112,192,292,592 that admit a non-solvable doubly transitive line-sized orbit are completely classified.

Quadratic Extensions of Flag-transitive Planes

In this article, the finite translation planes which are quadratic extensions of flag-transitive subplanes are classified as either Desarguesian or Hall.

Construction of Divisible Designs from Translation Planes

We construct some classes of divisible designs from finite translation planes of dimension two and three overGF(q),qa prime power, admittingSL(2,q) as a collineation group.

Some Multiply Derived Translation Planes withSL(2,5) as an Inherited Collineation Group in the TranslationComplement

Finite translation planes having a collineation group isomorphic to SL(2,5) occur in many investigations on minimal normal non-solvable subgroups of linear translation complements. In this paper, we are looking for multiply derived translation planes of the desarguesian plane which have an inherited linear collineation group isomorphic to SL(2,5). The Hall plane and some of the planes discovered by Prohaska [10], see also [1], are translation planes of this kind of order q², provided that q is odd and either q² ≡ 1 mod 5 or q is a power of 5. In this paper the case q² ≡ -1 mod 5 is considered and some examples are constructed under the further hypothesis that either q ≡ 2 mod 3, or q ≡ 1 mod 3 and q ≡ 1 mod 4, or q ≡ -1 mod 4, 3 n q and q ≡ 3,5 or 6 mod 7. One might expect that examples exist for each odd prime power q. But this is not always true according to Theorem 2.

Translation planes of large dimension admitting nonsolvable groups

In this article, the question is considered whether there exist finite translation planes with arbitrarily small kernels admitting nonsolvable collineation groups. For any integerN, it is shown that there exist translation planes of dimension >N and orderq3 admittingGL(2,q) as a collineation group.

Some finite translation planes arising from A6-1nvariant ovoids of the Klein quadric

Some finite translation planes arising from A6-1nvariant ovoids of the Klein quadric

Inherited Groups and Kernels of Derived Translation Planes

When an affine plane is converted to another plane by derivation, the point permutations which act as collineations of both planes form the inherited group. The full group can be larger than the inherited group. For finite translation planes in which some of the Baer subplanes involved are not vector spaces over the kernel of the original plane then the full collineation group of the derived plane is the inherited group provided the order of the plane is greater than 16.

Correction to ?Two-groups on finite translation planes?

Correction to ?Two-groups on finite translation planes?

Simple groups acting on translation planes

We discuss the possibility of finite simple groups acting as collineation groups on finite translation planes of odd order with special attention paid to the sporadic simple groups. We assume such a group acts irreducibly (in the vector space sense) on the plane. It is shown that if the characteristic of the plane does not divide the order of the group, then the group cannot be one of eleven sporadic simple groups. Also, if one of the Mathieu groups acts irreducibly on a finite translation plane then it is either M11 or M23.

Translation complements of C-planes: (I)

Narayana Rao, Rodabaugh, Wilke and Zemmer constructed a new class of finite translation planes from exceptional near-fields described by Dickson and Zassenhaus. These planes referred to as C-planes are not coordinatized by the generalized André systems. In this paper we compute the translation complement of the C-plane corresponding to the C-system III–1. It is found that the translation complement is of order 6912 and it divides the set of ideal points into two orbits of lengths 2 and 48.