Extremal orthogonal arrays
Alexander L. Gavrilyuk, Sho Suda
TL;DR
The work investigates extremal orthogonal arrays within the Hamming association scheme by leveraging Delsarte t-design theory. By showing that an extremal OA yields a fission scheme with $2s-1$ or $2s$ classes and deriving its second eigenmatrix, the authors obtain a new distance-inequality framework and a necessary modular condition for existence of tight 3-designs in $H(n,q)$. The results connect extremal designs to coding-theoretic objects (e.g., Golay codes) and extend previous inequalities of Ionin–Shrikhande to the OA/Hamming setting. Overall, the paper advances structural constraints and feasibility criteria for extremal and tight designs in Q-polynomial association schemes, with implications for the existence and classification of tight 3-designs in Hamming schemes.
Abstract
It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs in Johnson association schemes) having exactly $s$ block intersection numbers, we call a Delsarte $(2s-1)$-design with degree $s$ extremal and study extremal orthogonal arrays, which are Delsarte designs in Hamming association schemes. It was shown by Delsarte that a $t$-design with degree $s$ and $t\geq 2s-2$ in a Hamming association scheme induces an $s$-class association scheme. We prove that an extremal orthogonal array gives rise to a fission scheme of the latter one, which has $2s-1$ or $2s$ classes. As a corollary, a new necessary condition for the existence of tight orthogonal arrays of strength $3$ is obtained. Furthermore, as a counterpart to a result of Ionin and Shrikhande, we prove an inequality for Hamming distances in extremal orthogonal arrays. The inequality is tight as shown by examples related to the Golay codes.