The dual code of small linear spaces

Let p be an odd prime and C denote the p-ary code generated by lines of a linear space (X, ) of order n and C1/ its orthogonal. In [5], the following conjecture was made: ture 1 If n < p2, then the minimum weight of C1/ is at least 2n. s 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 < p2, we prove that words in its dual code of weight less than In must have at least four distinct coefficients in its expression. thod is to study the induced linear space structure on the support of a word of minimum weight of C1/ (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, ) is a linear space without a fixed number of lines through every point, or if it is a partial linear space. 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.