The dual code of small linear spaces

Abstract

Abstract Let p be an odd prime and C Download : Download full-size image denote the p-ary code generated by lines of a linear space (X, Download : Download full-size image ) of order n and C1/ Download : Download full-size image its orthogonal. In [5], the following conjecture was made: Conjecture 1 If n < p2, then the minimum weight of C1/ Download : Download full-size image is at least 2n. In this article we prove the conjecture for n = 2p. Our results imply that a projective plane of order 2p is tame at p, if it exists. We recall that the proof of the non-existence of a projective plane of order 10 involves ruling out words of certain weight in its binary dual code. Therefore this result, while partially answering [1, Remark 2, p. 237], may shed some light on the general case. For a general linear space of order Our method is to study the induced linear space structure on the support of a word of minimum weight of C1/ Download : Download full-size image (the ‘small linear spaces’). However, unlike in [5], our bound on the number of points is not ‘sufficiently away’ from 2n which makes proofs technically harder. This subtlety is also expressed in Examples 1 and 2 which show that the statement of the Conjecture 1 is not true if (X, Download : Download full-size image ) is a linear space without a fixed number of lines through every point, or if it is a partial linear space. It is natural to ask : what happens if the order of the linear space equals 3p or some such small multiple of p? It is reasonably clear to us that similar methods will give the proof of the conjecture in case n = 3p. Since the case n = 2p illustrates the main points of our approach, we have concentrated only on that case here.