On Characterizing Designs by Their Codes

Abstract

We observe that the finite regular generalized polygons are characterized by their codes (over any field!): they are the only generalized polygons (with given parameters) which maximise the number of minimum weight words in the dual code. We conjecture that an analogous characterization (over $${\mathbb{F}}_{p}$$ , p the correct prime) holds at least for the point-line designs of a finite dimensional projective space over a prime field. In support of this conjecture, we present a weaker coding theoretic characterization of these design in terms of the notion of “large clubs” introduced here. Along the way, we also prove a combinatorial characterization of the point-line designs of all finite projective spaces: apart from projective planes, these are the only Steiner 2-designs having as many hyperplanes as points. A similar characterization of the desarguesian projective plane among projective planes of a prime order is not expected, except perhaps rather vacuously. However, we conjecture that the prime order desarguesian planes are characterized by maximising the number of words of the second minimum weight in their dual codes. We state a conjecture, on small linear spaces of prime order, whose validity is shown to imply this conjecture for projective planes.