  We investigate the randomized and quantum communication complexity of the
Hamming Distance problem, which is to determine if the Hamming distance between
two n-bit strings is no less than a threshold d. We prove a quantum lower bound
of \Omega(d) qubits in the general interactive model with shared prior
entanglement. We also construct a classical protocol of O(d \log d) bits in the
restricted Simultaneous Message Passing model, improving previous protocols of
O(d^2) bits (A. C.-C. Yao, Proceedings of the Thirty-Fifth Annual ACM Symposium
on Theory of Computing, pp. 77-81, 2003), and O(d\log n) bits (D. Gavinsky, J.
Kempe, and R. de Wolf, quant-ph/0411051, 2004).
