In the theory of orthogonal arrays, an orthogonal array (OA) is called schematic if its rows form an association scheme with respect to Hamming distances. In this paper, we study the Hamming distances of any two rows in an OA, construct some schematic OAs of strength two and give the positive solution to the open problem for classifying all schematic OAs. Some examples are given to illustrate our methods.

This paper discusses the problem of reducing the number of keys required to authenticate a source state (plaintext), as well as introducing a new way of constructing authentication codes. The construction uses association schemes, which are well-defined schemes in combinatorial design theory. The association scheme on the message (ciphertext) space is established by defining two relations between any two messages: The 1st relation is when the two messages do not share a common key, and the 2nd relation is when they do. Using association schemes of the triangular and group divisible types, we are able to reduce the number of keys.