Properties of Codes with Two Homogeneous Weights

TL;DR

This paper generalizes properties of two-weight codes from finite fields to finite Frobenius rings, establishing their weight structure and connection to strongly regular graphs, and explores existence questions for such codes.

Contribution

It extends the characterization of two-weight codes from finite fields to Frobenius rings and provides new proofs and insights into their structure and associated graphs.

Findings

Two-weight codes over Frobenius rings have weights related by a divisor of the code order.

Proper regular projective two-weight codes yield strongly regular graphs.

The paper offers new existence results for two-weight codes over finite rings.

Abstract

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.