Moebius-Walsh correlation bounds and an estimate of Mauduit and Rivat

We establish small correlation bounds for the Moebius function and the Walsh system, answering affirmatively a question posed by G.Kalai [Ka]. The argument is based on generalizing the approach of Mauduit and Rivat [M-R] in order to treat Walsh functions of `large weight', while the `small weight' case follows from recent work due to B.Green [Gr]. The conclusion is an estimate uniform over the full Walsh system. A similar result also holds for the Liouville function.