A nondeterministic space-time tradeoff for linear codes

We are interested in proving exponential lower bounds on the size of nondeterministic D-way branching programs computing functions f: Dn → {0, 1} in linear time, that is, in time at most kn for a constant k. Ajtai has proved such lower bounds for explicit functions over domains D of size about n, and Beame, Saks and Thathachar for functions over domains of size about 22 k. We prove an exponential lower bound 2Ω(n/c k) for an explicit function over substantially smaller domain D of size about 2k. Our function is a universal function of linear codes.