Translation Planes Of Order Research Articles

The dimension of a subplane of a translation plane

It is shown that the commutative binary Knuth semifield planes of order $2^{n}$, for $n=5k~$ or $7k$ and $k$ odd, their transposes and transpose-duals admit subplanes of order $2^{2}$. In addition, many of the Kantor commutative semifield planes of order $2^{5k}$ or $2^{7k}$ also admit subplanes of order $4$. Furthermore, a large number of maximal partial spreads of order $p^{k}$ and deficiency at least $p^{k}-p^{k-1}$ or translation planes of order $p^{k}$ are constructed using direct sums of matrix spreads sets of different dimensions. Given any translation plane $\pi_{0}$ of order $p^{d}$, there is either a proper maximal partial spread of order $p^{c+d}$ whose associated translation net contains a subplane of order $p^{d}$ isomorphic to $\pi_{0}$ or there is a translation plane of order $p^{c+d}$ admitting a subplane of order $p^{d}$. Other than the semifield planes mentioned above and a few sporadic planes of even order, there are no other known translation planes of order $p^{c+d}$ admitting a subplane of order $p^{d}$, where $d$ does not divide $c$.

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and πt, the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that πt is flag-transitive, πt is not isomorphic to π, and πt is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S2b and S2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and πt (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and πt in some order.

Further translation planes of order 192 admitting SL(2,5), obtained by nest replacement

We construct three translation planes of order 192 admitting SL(2,5), obtained by replacement of 24-nests of reguli in PG(3,19).

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.

Mixed partitions and related designs

We define a mixed partition of ? = PG(d, q r ) to be a partition of the points of ? 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 Andre planes, thereby providing a geometric representation of an infinite family of generalized Andre 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 ).

Derivable André and generalized André Planes

There are André and generalized André planes of order q 2n that are shown to bederivable. The derived planes are not generalized André planes and admit a Baer group B of order (q n − 1)/(q − 1) x a cyclic group Z of order (q 2n − 1) / (q - 1) A characterizationis given of translation planes of order q 2n admitting B × Z, where B is Baer.

A translation plane of order 192 admitting SL(2,5), obtained by 12-nest replacement

The problem of determining two-dimensional translation planes admitting SL(2,5) in the translation complement has been intensively studied. Most examples arise from multiple-derivation of a Desarguesian plane. In this paper, we construct two translation planes of order 192 admitting SL(2,5), one of which is obtained by 12-nest replacement.

The translation planes of order 81 that admit SL(2,5), generated by affine elations

The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes, of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new.

Classical mixed partitions

Through a method given in Hirschfeld and Thas (General Galois Geometries, Oxford University Press, Oxford, 1991), a mixed partition of PG(2 n −1, q 2 ) can be used to construct a (2 n −1)-spread of PG(4 n −1, q ) and, hence, a translation plane of order q 2 n . A mixed partition in this case is a partition of the points of PG(2 n −1, q 2 ) into PG( n −1, q 2 )'s and PG(2 n −1, q )'s which we call Baer subspaces. In this paper, we completely classify the mixed partitions which generate regular spreads and, hence, can be classified as classical .

Mixed partitions of PG(3, q2)

A mixed partition of PG(2 n−1, q 2) is a partition of the points of PG(2 n−1, q 2) into ( n−1)-spaces and Baer subspaces of dimension 2 n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2 n−1, q 2) can be used to construct a (2 n−1)-spread of PG(4 n−1, q) and hence a translation plane of order q 2 n . In this paper, we provide several new examples of such mixed partitions in the case when n=2.

Classical mixed partitions

Through a method given in Hirschfeld and Thas (General Galois Geometries, Oxford University Press, Oxford, 1991), a mixed partition of PG(2 n−1, q 2) can be used to construct a (2 n−1)-spread of PG(4 n−1, q) and, hence, a translation plane of order q 2 n . A mixed partition in this case is a partition of the points of PG(2 n−1, q 2) into PG( n−1, q 2)'s and PG(2 n−1, q)'s which we call Baer subspaces. In this paper, we completely classify the mixed partitions which generate regular spreads and, hence, can be classified as classical.

Doubly β-Derived Translation Planes

Existence and uniqueness of a doubly β-derived translation plane of order 49 are proved. Furthermore, we give a complete classification of those translation planes of order 49 which can be obtained from the desarguesian plane Π of order 49 by a mixed double derivation, namely by applying a β-derivation on Π and a classical derivation (also called Ostrom's derivation or δ-derivation) on βΠ.

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order $ p^t $ which admits two homology groups of order $ (p^t-1)/3 $ must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized Andre plane. A list of the possibly exceptional orders is determined.

A note on line-Baer subspace partitions of PG(3, 4)

We consider the partitioning of PG (3,4) into two types of objects, lines and Baer sub-3-spaces. Any such mixed partition gives rise to a spread of PG(7,2) (and hence a projective plane of order 16) via a construction technique given in [6]. The author has used the software package Magma to determine all such mixed partitions up to equivalence. It turns out that all of the translation planes of order 16 arise from one of the mixed partitions.

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.

Some inherited maximal arcs in derived dual translation planes

In 1974 J. A. Thas constructed a class of maximal arcs in certain translation planes of order q(2). In this paper a new class of maximal arcs is constructed in certain derived dual translation planes that are inherited from the duals of the Thas maximal arcs. It is noted that some (but not all) of the maximal arcs are isomorphic to a class constructed by the author.

A characterisation of thas maximal arcs in translation planes of square order

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of order q2. We characterise these as being exactly those (non-trivial) maximal arcs that are stabilised by an homology of order q− 1.

On the Sherk plane

In this paper, we characterize the Sherk plane as the only non-semifield translation plane of order 27 that admits an abelian group of order 27 in the translation complement.

Translation planes of order 27

Translation planes of order 27 are classified. Various invariants play an important role in a computer search.

Projective Planes of Order q whose Collineation Groups have Order q 2

Translation planes of order q are constructed whose full collineation groups have order q 2.