Research
Selected results, public records and work that remains open.
Certified Small Signed Difference Sets Census
A computer-assisted census of 68 cases listed as open in a frozen upstream snapshot, for abelian groups of order at most 36.
- 16 explicit constructions and 52 nonexistence results.
- For order 32, a signed (32,20,4) difference set exists exactly for the six noncyclic abelian groups.
- No abelian group of order 36 admits a signed (36,29,4) difference set.
Publication status: v1.1 was released publicly on GitHub on 23 September 2026, with a revised manuscript and reproducibility package. Fabian Arévalo independently reviewed the work; Daniel Gordon encouraged journal submission. No journal acceptance or submission has been identified.
Read the v1.1 release. Open the v1.0 Zenodo archive. Project details.
The ten entries described as new are supported by a bounded prior-art search; that search does not establish universal novelty.
Boolean Circuit Discovery
Reproducible AES S-box circuit transformations with 29 AND gates, keeping gate count and depth as separate objectives.
- 158 XOR gates at depth 30.
- 159 XOR gates at depth 29.
- An earlier 161-XOR circuit was independently verified by René Peralta; that verification does not automatically cover the later variants.
Public record: the NIST Circuit Complexity list credits Nicolas and lists the 158-XOR and 159-XOR transformations. A Zenodo record archives the work. No journal publication is established by the available record.
Open investigations
- C(16,7,3): a 23-block candidate covers 558 of 560 triples. The general covering problem remains unresolved.
- Proto-Elamite: a reimplementation and arithmetic-constraint exploration on a published corpus. No decipherment is claimed.