This note reports the result of a computer search which shows that no 19-point con guration can be extended to a complete nite projective plane of order 10. Previous work has shown that such a plane, if it exists, must contain 24,675 19-point con gurations. Together, these results imply the non-existence of a projective plane of order 10. This note presents a brief summary of the previous work and gives some details of the methodology which are not covered by earlier publications. It also argues that, even when the possibility of undetected software or hardware errors is taken into account, the probability is very small that an undiscovered plane of order 10 is missed by all the computer searches.

The symmetric class-regular (4, 4)-nets having a group of bitranslations G of order four are enumerated up to isomorphism. There are 226 nets with G ∼ = Z 2 × Z 2 , and 13 nets with G ∼ = Z 4 . Using a (4, 4)-net with full automorphism group of smallest order, the lower bound on the number of pairwise non-isomorphic affine 2-(64, 16, 5) designs is improved to 21,621,600. The classification of classregular (4, 4)-nets implies the classification of all generalized Hadamard matrices (or difference matrices) of order 16 over a group of order four up to monomial equivalence. The binary linear codes spanned by the incidence matrices of the nets, as well as the quaternary and Z 4 -codes spanned by the generalized Hadamard matrices are computed and classified. The binary codes include the affine geometry [64,16,16] code spanned by the planes in AG(3, 4) and two other inequivalent codes with the same weight distribution. These codes support non-isomorphic affine 2-(64, 16, 5) designs that have the same 2-rank as the classical affine design in AG(3, 4), hence provide counter-examples to Hamada's conjecture. Many of the F 4 -codes spanned by generalized Hadamard matrices are self-orthogonal with respect to the Hermitian inner product and yield quantum error-correcting codes, including some codes with optimal parameters.

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