A construction of [formula omitted] by nonlinear functions and some classification for [formula omitted

Abstract

In the case of s = 2 , Seiden and Zemach [1966. On orthogonal arrays. Ann. Math. Statist. 37, 1355–1370] classified all OA ( 2 t λ , t + 1 , 2 , t ) for any λ . Fujii et al. [1989. Classification of two symbol orthogonal arrays of strength t, t + 3 constraints and index 4, II. SUT J. Math. 25, 161–177] classified OA ( 2 t + 1 , t + 2 , 2 , t ) and OA ( 2 t + 2 , t + 3 , 2 , t ) . Recently, Stufken and Tang [2006. Complete enumeration of two-level orthogonal arrays of strength d with d + 2 constraints. Preprint] completely enumerated OA ( 2 t λ , t + 2 , 2 , t ) . Moreover, in the case of s = 3 , Hedayat et al. [1997a. On the maximum number of factors and the enumeration of 3-symbol orthogonal arrays of strength 3 and index 2. J. Statist. Plann. Inference 58, 43–63; 1997b. On the construction and existence of orthogonal arrays with three levels and indexes 1 and 2. Ann. Statist. 25, 2044–2053] showed that all OA ( 3 t , t + 1 , 3 , t ) are isomorphic and linear. Hedayat et al. [1997a. On the maximum number of factors and the enumeration of 3-symbol orthogonal arrays of strength 3 and index 2. J. Statist. Plann. Inference 58, 43–63] classified all OA ( 2 · 3 3 , 5 , 3 , 3 ) . In this paper, we treat the case of s ⩾ 4 . Firstly, we show a construction theorem of OA ( s t , t + 1 , s , t ) by utilizing nonlinear functions over GF ( s ) , where s = p l , p a prime, and l a multiple of p. Secondly, we classify OA ( 4 t , t + 1 , 4 , t ) for t = 2 , 3 and 4 by utilizing their functional representations.