Properties of codes with two homogeneous weights

Delsarte showed that for any projective linear code over a finite field GF(p(r)) with two nonzero Hamming weights w(1) < w(2) there exist positive integers u and s such that w(1) = p(s)u and w(2) = 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 regular projective linear code C over a finite Frobenius ring with two integral nonzero homogeneous weights w(1) < w(2) there is a positive integer d, a divisor of vertical bar C vertical bar, and positive integer u such that w(1) = du and w(2) = d(u + 1). This gives a new proof of the known result that any such code yields a strongly regular graph. We apply these results to existence questions on two-weight codes. (C) 2012 Elsevier Inc. All rights reserved.