Symmetric bilinear forms over finite fields of even characteristic

Let S m be the set of symmetric bilinear forms on an m-dimensional vector space over GF ( q ) , where q is a power of two. A subset Y of S m is called an ( m , d ) -set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four. An upper bound on the size of ( m , d ) -sets is derived, and in certain cases, the rank distance distribution of an ( m , d ) -set is explicitly given. Constructions of ( m , d ) -sets are provided for all possible values of m and d.