Exponential numbers of two-weight codes, difference sets and symmetric designs

The purpose of this paper is to obtain exponential lower bounds on the numbers of non-isomorphic linear codes or symmetric designs of certain types. This will be accomplished using a famil iar--even mundane-ob jec t related to the desarguesian affine plane AG(2, q"). Namely, let V be a 2-dimensional vector space over GF(q") , and let A be its set of l-spaces. We will use subsets £ of A to define a code C:~ and, when q = 2, a difference set U~ and a symmetric design D~. As in [5], we will then use Sylow's Theorem in order to deal with isomorphisn,, questions.