Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets (Q1003652)

H. Cohn et. al. proposed an association scheme of 64 points in \(R^{14}\) which is conjectured to be a universally optimal code. Authors showed that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. It is explained this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.