Extremal orthogonal arrays

TL;DR

This paper investigates extremal orthogonal arrays within Hamming association schemes, establishing their structural properties, deriving a new necessary condition for tight arrays, and providing bounds on Hamming distances with examples from Golay codes.

Contribution

It introduces the concept of extremal orthogonal arrays, shows their relation to fission schemes, and derives new conditions and bounds for their existence and properties.

Findings

Extremal orthogonal arrays induce fission schemes with 2s-1 or 2s classes.

A new necessary condition for the existence of tight orthogonal arrays of strength 3.

An inequality for Hamming distances in extremal orthogonal arrays, tightness demonstrated by Golay code examples.

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.