On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes

摘要

In this paper we present the complete classification of triplanes (71,15,3) admitting an action of the cyclic automorphism group of order six. Up to isomorphism there are 146 such triplanes and these are all triplanes of order 12 known up to now. Further, we analyze binary and ternary codes spanned by the incidence matrices of triplanes (71,15,3) and their residual designs. The constructed binary codes are self-complementary, and the ternary codes are self-orthogonal. In addition, we study ternary self-orthogonal codes constructed-from the orbit matrices for Z(3) acting on the 146 symmetric 2-(71, 15, 3) designs. Some of the obtained codes have minimum distance one or two less than the best known codes with the same length and dimension. Finally, we discuss k-geodetic graphs from the symmetric (71, 15, 3) designs and their residual and derived designs.