On extremal orthogonal arrays

Abstract: An orthogonal array with parameters $(N,n,q,t)$ ($OA(N,n,q,t)$ for short) is an $N\times n$ matrix with entries from the alphabet $\{1,2,...,q\}$ such that in any its $t$ columns, all possible row vectors of length $t$ occur equally often. Rao showed the following lower bound on $N$ for $OA(N,n,q,2e)$: \[ N\geq \sum_{k=0}^e \binom{n}{k}(q-1)^k, \] and an orthogonal array is said to be complete or tight if it achieves equality in this bound. It is known by Delsarte (1973) that for complete orthogonal arrays $OA(N,n,q,2e)$, the number of Hamming distances between distinct two rows is $e$. One of the classical problems is to classify complete orthogonal arrays.

We call an orthogonal array $OA(N,n,q,2e-1)$ extremal if the number of Hamming distances between distinct two rows is $e$. In this talk, we review the classification problem of complete orthogonal arrays with our contribution to the case $t=4$ and show how to extend it to extremal orthogonal arrays. Moreover, we give a result for extremal orthogonal arrays which is a counterpart of a result in block designs by Ionin and Shrikhande in 1993.

combinatoricsnumber theory

Audience: researchers in the topic

Lethbridge number theory and combinatorics seminar