Ring geometries, Two-Weight Codes and Strongly Regular Graphs

TL;DR

This paper extends the classical correspondence between two-weight codes and strongly regular graphs from finite fields to finite Frobenius rings, providing new constructions and infinite families of such graphs.

Contribution

It generalizes the known relationship to ring-linear codes and introduces methods for constructing two-weight codes via ring geometries.

Findings

Two-weight codes over Frobenius rings correspond to strongly regular graphs.

Constructed infinite families of strongly regular graphs from ring geometries.

Abstract

It is known that a linear two-weight code $C$ over a finite field $\F_q$ corresponds both to a multiset in a projective space over $\F_q$ 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 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.