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

We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combin. 23 (2002) 529–538] 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 pmpm where pp is a prime, and m≥1m≥1. This gives words of weight 2pm+1−pm−1p−1 in the dual of the pp-ary code of the desarguesian plane of order pmpm, 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 [Gábor Korchmáros, Francesco Mazzocca, On (q+t)(q+t)-arcs of type (0,2,t)(0,2,t) in a desarguesian plane of order qq, Math. Proc. Cambridge Philos. Soc. 108 (1990) 445–459].