A qxn array with entries from {0, 1 ..... q-l} is said to form a difference matrix if the vector difference (modulo q) of each pair of columns consists of a permutation of {0, 1 ..... q -1 }; this definition is inverted from the more standard one to be found, e.g., in Colbourn and de Launey (1996). The following idea generalizes this notion: Given an appropriate A C_{-l, 1} t, a 2q Γ n array will be said to form a (t,q,2,A) sign-balanced matrix if for each choice Q,C2 ..... Ct of t columns and for each choice e = (el ..... et) E A of signs, the linear combination ~=1 ejCj contains (rood q) each entry of {0, 1 ..... q -1} exactly 2 times. We consider the following extremal problem in this paper: How large does the number k = k(n, t, q, 2, A) of rows have to be so that for each choice of t columns and for each choice (et ..... et) of signs in A, the linear
combination ~_~ ejC j contains each entry of {0, 1 ..... q -1) at least 2 times? We use probabilistic metho~l's~ in particular the Lovhsz local lemma and the Stein-Chen method of Poisson approximation to obtain general (logarithmic) upper bounds on the numbers k(n,t,q,2, A), and to provide Poisson approximations for the probability distribution of the number W of deficient sets of t columns, given a random array. It is proved, in addition, that arithmetic modulo q yields the smallest array --in a sense to be described. (~