Symmetric bilinear forms over finite fields of even characteristic

Let Sm 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 Sm 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.