Bilinear forms over a finite field,with applications to coding theory

Let Ω be the set of bilinear forms on a pair of finite-dimensional vector spaces over GF(q). If two bilinear forms are associated according to their q-distance (i.e., the rank of their difference), then Ω becomes an association scheme. The characters of the adjacency algebra of Ω, which yield the MacWilliams transform on q-distance enumerators, are expressed in terms of generalized Krawtchouk polynomials. The main emphasis is put on subsets of Ω and their q-distance structure. Certain q-ary codes are attached to a given $\text{X ? Ω}$; the Hamming distance enumerators of these codes depend only on the q-distance enumerator of X. Interesting examples are provided by Singleton systems $\text{X ? Ω}$, which are defined as t-designs of index 1 in a suitable semilattice (for a given integer t). The q-distance enumerator of a Singleton system is explicitly determined from the parameters. Finally, a construction of Singleton systems is given for all values of the parameters.