Spreads, Translation Planes and Kerdock Sets. I

In an orthogonal vector space of type $\Omega ^ + ( 4n,q )$, a spread is a family of $q^{2n - 1} + 1$ totally singular $2n$-spaces which induces a partition of the singular points; these spreads are closely related to Kerdock sets. In a $2m$-dimensional vector space over $GF ( q )$, a spread is a family of $q^m + 1$ subspaces of dimension m which induces a partition of the points of the underlying projective space; these spreads correspond to affine translation planes. By combining geometric, group theoretic and matrix methods, new types of spreads are constructed and old examples are studied. New Kerdock sets and new translation planesare obtained having various interesting properties.