Orthogonal arrays obtainable as solutions to linear equations over finite fields

AbstractLet T(n x N) be a matrix with elements from the set of s integers {0,1…, s −1} Then T is said to be an orthogonal array (OA) with s symbols, n rows, N columns, strength p, and index λ(=Ns−p) if and only if, for every p x N submatrix T0 of T, each of the sp possible p x 1 column vectors occur as a column of T0 exactly λ times. Now, let s be a prime number or a power of a prime number, and let GF(s) be the finite field with s elements. Consider the equation At = ci(i = 1,…, f) over GF(s), where A (r x n) is a matrix of rank r, and ci(r x 1) are vectors (not necessarily distinct). For i = 1,…,f let Ti (n x sn−r) be a matrix whose columns represent the sn−r distinct solutions of the equation At= ci. Let T be the (n x fsn−r) matrix given by T = [T1:T2:⋯:Tf]. In this paper, we study necessary and sufficient conditions on the pair (A, C) (where C = [c1,…,cf]) for T to be an orthogonal array of strength p