Ring geometries, two-weight codes, and strongly regular graphs

Abstract

It is known that a projective linear two-weight code C over a finite field $${\mathbb{F}}_q$$ corresponds both to a set of points in a projective space over $${\mathbb{F}}_q$$ that meets every hyperplane in either a or b points for some integers a < b, and to a strongly regular graph whose vertices may be identified with the codewords of C. Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homogeneous weights and sets of points in an associated projective ring geometry. We will introduce regular projective two-weight codes over finite Frobenius rings, we will show that such a code gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. All these examples yield infinite families of strongly regular graphs with non-trivial parameters.