TY - JOUR

T1 - An upper bound for the minimum weight of the dual codes of desarguesian planes

AU - Mavron, V. C.

AU - McDonough, Thomas

AU - Key, Jennifer D.

N1 - J.D.Key, T.P.McDonough and V.C.Mavron, An upper bound for the minimum weight of the dual codes of desarguesian planes. European Journal of Combinatorics, Volume 30 Issue 1, January, 2009, pp. 220-229.

PY - 2009/1

Y1 - 2009/1

N2 - We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combinatorics 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 p^m where p is a prime, and m≥1. This gives words of weight 2p^m+1-(p^m-1)/(p-1) in the dual of the p-ary code of the desarguesian plane of order p^m, 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)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Phil. Soc. 108 (1990) 445–459].

AB - We show that a construction described in [K.L. Clark, J.D. Key, M.J. de Resmini, Dual codes of translation planes, European J. Combinatorics 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 p^m where p is a prime, and m≥1. This gives words of weight 2p^m+1-(p^m-1)/(p-1) in the dual of the p-ary code of the desarguesian plane of order p^m, 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)-arcs of type (0,2,t) in a desarguesian plane of order q, Math. Proc. Cambridge Phil. Soc. 108 (1990) 445–459].

U2 - 10.1016/j.ejc.2008.01.003

DO - 10.1016/j.ejc.2008.01.003

M3 - Article

SN - 0195-6698

VL - 30

SP - 220

EP - 229

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

IS - 1

ER -