Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

摘要

All Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2_(63,31,15) designs having line spreads by Rahilly'_s construction A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291_303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(3, p~m) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3, 4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71-87].