On tight $4$-designs in Hamming association schemes

TL;DR

This paper fully classifies tight 4-designs in Hamming association schemes by constructing associated schemes and analyzing their intersection numbers, resolving an open problem since 1979.

Contribution

It completes the classification of tight 4-designs in Hamming schemes by proving non-existence in all open cases through new association scheme analysis.

Findings

No tight 4-designs exist in the open cases

Constructed association schemes linked to tight 4-designs

Analyzed triple intersection numbers to establish non-existence

Abstract

We complete the classification of tight $4$-designs in Hamming association schemes $H(n,q)$, i.e., that of tight orthogonal arrays of strength $4$, which had been open since a result by Noda (1979). To do so, we construct an association scheme attached to a tight $4$-design in $H(n,q)$ and analyze its triple intersection numbers to conclude the non-existence in all open cases.