Given a finite abelian group G of order v and integers k and lambda, determine whether there exists a function x:G→-1,0,1 such that ∑_g∈ Gx(g)^2=k and, for every nonzero h∈ G, ∑_g∈ Gx(g)x(g+h)=lambda. Equivalently, determine which finite abelian groups admit a signed difference set with prescribed parameters (v,k,lambda).

Open group-specific existence problems for signed difference sets

The problem asks which finite groups admit signed difference sets with prescribed parameters. Earlier work established general constructions and computational examples, but did not settle the full collection of group-specific cases.

Known results

General existence results and computational searches for signed difference sets (2022–2023).
An explicit noncyclic example with parameters $(18,13,4)$ in $\mathbb{Z}_{2}\times\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ was reported in earlier work.

August 2026 obstructions and constructions

A new preprint introduces four nonexistence obstructions, an infinite construction family, and an explicit $(125,28,3)$ signed difference set in $C_{5}^{3}$ . It claims to resolve three previously open cases and substantially reduce the search space, while many group-specific cases remain open; these claims are unverified here.

Current status (as of August 2026): Three cases are claimed resolved by new obstructions and constructions, but the claim is unverified and many group-specific existence problems remain open.

Open group-specific existence problems for signed difference sets

Primary source

Additional references

Progress summary

Solutions 0