Abstract
Using multi-party communication techniques, we prove that depth-3 circuits with a threshold gate at the top, arbitrary symmetric gates at the next level, and fan-in k MODm gates at the bottom need exponential size to compute the k-wise inner product function of Babai, Nisan and Szegedy, where m is an odd positive integer satisfying \( m \equiv k \bmod 2m \). This is one of the rare lower-bound results involving MODm gates with non-prime power moduli.¶Exponential gap theorems are also presented between the multi-party communication complexities of closely related functions.
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