Additive bases of vector spaces over prime fields

Let Z; be the n-dimensional linear space over the prime field Z,. An additive basis of Z; is a multiset {x,, . . . . x,} c Z;, such that any XE Z; is representable as a Ckl combination of the x;s. Let j(p, n) denote the mmima1 integer t, such that for any t (linear) bases B,, . . . . B, of Z;, the union (with repetitions) vi= r Bi forms an additive basis of Z;. The problem of determining or estimating f(p, n), besides being interesting in its own right, is naturally motivated by the study of universal