An upper bound for the minimum weight of the dual codes of desarguesian planes

We show that a construction described in Clark, Key and de Resmini [9] of small-weight words in the dual codes of finite translation planes can be extended so that it applies to projective and affine desarguesian planes of any order p where p is a prime, and m ≥ 1. This gives words of weight 2p + 1 − p −1 p−1 in the dual of the p-ary code of the desarguesian plane of order p, and provides an improved upper bound for the minimum weight of the dual code. The same will apply to a class of translation planes that this construction leads to; these belong to the class of André planes. We also found by computer search a word of weight 36 in the dual binary code of the desarguesian plane of order 32, thus extending a result of Korchmáros and Mazzocca [19].