On two-weight codes

We consider q-ary (linear and nonlinear) block codes with exactly two distances: d and d+δ. We derive necessary conditions for existence of such codes (similar to the known conditions in the projective case). In the linear (but not necessary projective) case, we prove that under certain conditions the existence of such linear 2-weight code with δ>1 implies the following equality of greatest common divisors: (d,q)=(δ,q). Upper bounds for the maximum cardinality of such codes are derived by linear programming and from few-distance spherical codes. Tables of lower and upper bounds for small q=2,3,4 and qn