2 research outputs found
Uniqueness of Optimal Mod 3 Circuits for Parity
We prove that the quadratic polynomials modulo
with the largest correlation with parity are unique up to
permutation of variables and constant factors. As a consequence of
our result, we completely characterize the smallest
MAJ~ circuits that compute parity, where a
MAJ~ circuit is one that has a
majority gate as output, a middle layer of MOD gates and a
bottom layer of AND gates of fan-in . We
also prove that the sub-optimal circuits exhibit a stepped behavior:
any sub-optimal circuits of this class that compute parity
must have size at least a factor of times the
optimal size. This verifies, for the special case of ,
two conjectures made
by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~ circuits for any odd . The correlation
and circuit bounds are obtained by studying the associated
exponential sums, based on some of the techniques developed
by Green (JCSS, 2004). We regard this as a step towards
obtaining tighter bounds both for the quadratic
case as well as for
higher degrees