A nondeterministic space-time tradeoff for linear codes

We consider nondeterministic D-way branching programs computing functions f : D → {0, 1} in time at most kn. In the boolean case, where D = {0, 1}, no exponential lower bounds are known even for k = 2. Ajtai has proved such lower bounds for explicit functions over domains D of size at least n, and Beame, Saks and Thathachar for explicit functions over domains of size at least 2 k . In this note we prove such a lower bound for an explicit function f : D → {0, 1} over substantially smaller domain of size about 2. Our function f(Y, x) has n +n variables, the first n of which are arranged in an n×n matrix Y . The variables take their values in the domain D = GF (q) for a prime power q about 2, and f(Y, x) = 1 iff the vector x is orthogonal over GF (q) to all rows of Y .