Orthomorphisms and the construction of projective planes

Abstract

Abstract. We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field GF (9) as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four selfdual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the "sporadic" (not translation) plane of Mathon.

Isomorphism criterion for monomial graphs

“…Given q and two functions f and g, when are G q ð f Þ and G q ðgÞ isomorphic? The question has already been shown to be hard for 4-cycle free graphs, since it is related to the question of isomorphism of finite projective planes, see [4]. To our knowledge, the only non-trivial case for the isomorphism problem which has been partially settled is the case when both f and g can be represented by monomials.…”

Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

“…The second motivation relates to incidence geometry, which we briefly discuss here. In two dimensions, it is known (see Dmytrenko [2] and Lazebnik and Thomason [14]) that every graph Γ Fq (f ) with girth greater than four can be completed to a projective plane of order q (although not all projective planes of order q can be constructed in this way). The three-dimensional analogue is motivated by the construction of generalized quadrangles because when q is even, there exist monomial (and non-monomial) graphs Γ Fq (f 2 , f 3 ) that can be used to construct non-isomorphic generalized quadrangles of order q.…”

“…We end this section with the following isomorphisms of the graph Γ F (f 2 , f 3 ), where F is a field; see, e.g., [14] (p. 3) or [11] (Proposition 2.2, p. 190) for proofs. First note that for a function f = f (X, Y ), we define…”

On monomial graphs of girth eight

“…It is easy to see that the graph Γ 2 is of order 2q 2 , q-regular and of girth six. See, e.g., [22] or Lazebnik and Thomason [18] for details.…”

“…For 4-cycle-free graphs G q (f 2 ) such extension is always possible and unique. One may try to use this fact to construct new projective planes, and many nonisomorphic finite projective planes can be obtained this way (see, e.g., [18,25]). If f 2 is a monomial, it is easy to argue (or see [9]) that every 4-cycle-free graph G q (f 2 ) is isomorphic to G q (xy), and the extended graph is the point-line incidence graph of the projective plane PG (2, q).…”