Properties of Codes with Two Homogeneous Weights

Delsarte showed that for any projective linear code over a finite field of characteristic p with two nonzero Hamming weights w1 < w2 there exist positive integers u and s such that w1 = (p^s)u and w2 = (p^s)(u+1). Moreover, he showed that the additive group of such a code has a strongly regular Cayley graph. Here we show that for any proper regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w1 < w2, there is a positive integer d, a divisor of the order of C, and positive integer u such that w1 = du and w2 = d(u+1). In doing so, we give a new proof of the known result that any proper regular projective two-weight code code yields a strongly regular graph. We apply these results to existence questions on two-weight codes.