Ring geometries, Two-Weight Codes and Strongly Regular Graphs
Abstract
It is known that a linear two-weight code C over a finite field q corresponds both to a multiset in a projective space over 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 multisets of points in an associated projective ring geometry. We will show that a two-weight code over a finite Frobenius ring gives rise to a strongly regular graph, and we will give some constructions of two-weight codes using ring geometries. These examples all yield infinite families of strongly regular graphs with non-trivial parameters.