Groups of Generalised Projectivities in Projective Planes of Odd Order

Abstract. The generalised projectivities (GP's) associated with projective planes of odd order are investigated. These are non-singular linear mappings over GF(2) defined from the binary codes of these planes. One case that is investigated in detail corresponds to the group of an affine plane- every point corresponds to a GP. It is shown how many collineations that fix the line at infinity point-wise can be constructed as a product of these GP's. 1. The Self-dual Code Since our groups of generalised projectivities will be constructed from mappings associated with the main self-dual code of a finite projective plane of odd order, let us recall some of the properties of this code; see [2] for further details. First, we have some notation. Let 1r be a finite projective plane of odd order q. Let P and L be the sets of points and lines respectively of 1r. Thus IPI = ILl = q2 + q + 1. If S is any set define Sf3 to be 0 if lSI is even, or 1 if lSI is odd. Also, if 9 E P or L let § be the set of q + 1 elements of P U L that are incident with g. Note that often we identify 9 E P U L with the set {g}. (This is not unusual in geometry.