Z4-kerdock codes, orthogonal spreads, and extremal euclidean line-sets

When m is odd, spreads in an orthogonal vector space of type O+(2m + 2,2) are related to binary Kerdock codes and extremal line-sets in 2m + 1 with prescribed angles. Spreads in a 2m-dimensional binary symplectic vector space are related to Kerdock codes over Z4 and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding Z4-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite m, of large numbers of Z4-Kerdock codes. They also produce new Z4-linear Kerdock and Preparata codes. 1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.