Ring geometries, Two-Weight Codes and Strongly Regular Graphs

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.

💡 Research Summary

The paper extends the classical correspondence between two‑weight linear codes over finite fields, projective multisets that intersect every hyperplane in exactly two possible cardinalities, and strongly regular graphs (SRGs). The authors replace the underlying field with a finite Frobenius ring R and investigate R‑linear codes that possess exactly two non‑zero homogeneous weights.

First, the algebraic setting is established. A finite Frobenius ring is a finite abelian group equipped with a non‑degenerate bilinear form, which guarantees the existence of a well‑behaved duality and a unit group U(R). The free R‑module V = R^k gives rise to a projective ring geometry PG(R, k); points are the non‑zero cyclic submodules ⟨v⟩, and hyperplanes correspond to proper submodules of codimension one. Unlike the field case, points may appear with multiplicities, reflecting the richer module structure.

Next, the authors define a two‑weight R‑linear code C ⊂ R^n. The homogeneous weight of a codeword x is the number of non‑zero coordinates counted with respect to the unit group of R; this weight takes only two distinct values w₁ < w₂ for all non‑zero codewords. By analysing the generator matrix G, they derive necessary and sufficient conditions for a code to be two‑weight: essentially G must be “regular” with respect to the ring’s unit action, and the image of G under the natural projection onto the residue field 𝔽_q must be a classical two‑weight code.

The crucial step is the construction of a multiset M of points in PG(R, k) from C. Each non‑zero codeword x determines a point