We show that there are non-desarguesian affine planes of order 16 for which the binary codes have vectors of minimum weight that are not the incidence vectors of lines. This is in contrast to the desarguesian case and answers an open question as to the nature of the minimum words of the code of a non-desarguesian affine plane. Further, we show that all the nontranslation planes of order 16 have hull of minimum weight smaller than 32, in fact containing words of weight 24. Most of these words of weight 24 yield words of weight 16 in the binary code of some affine plane of order 16 that are not the incidence vectors of affine lines. The search has also shown that all the non-desarguesian planes of order at most 16 are not tame. These results are also in contrast to what is known in the desarguesian case.
Minimum words of codes from affine planes / Ghinelli, Dina; DE RESMINI, Marialuisa; J. D., Key. - In: JOURNAL OF GEOMETRY. - ISSN 0047-2468. - STAMPA. - 91:(2008), pp. 43-51. [10.1007/s00022-008-2096-y]
Minimum words of codes from affine planes
GHINELLI, Dina;DE RESMINI, Marialuisa;
2008
Abstract
We show that there are non-desarguesian affine planes of order 16 for which the binary codes have vectors of minimum weight that are not the incidence vectors of lines. This is in contrast to the desarguesian case and answers an open question as to the nature of the minimum words of the code of a non-desarguesian affine plane. Further, we show that all the nontranslation planes of order 16 have hull of minimum weight smaller than 32, in fact containing words of weight 24. Most of these words of weight 24 yield words of weight 16 in the binary code of some affine plane of order 16 that are not the incidence vectors of affine lines. The search has also shown that all the non-desarguesian planes of order at most 16 are not tame. These results are also in contrast to what is known in the desarguesian case.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
