Abstract

We prove that the quadratic polynomials modulo $3$
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~$circ mbox{MOD}_3 circ {
m AND}_2$ circuits that compute parity, where a
MAJ~$circ mbox{MOD}_3 circ {
m AND}_2$ circuit is one that has a
majority gate as output, a middle layer of MOD$_3$ gates and a
bottom layer of AND gates of fan-in $2$. 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 $frac{2}{sqrt{3}}$ times the
optimal size. This verifies, for the special case of $m=3$,
two conjectures made
by Due~{n}ez, Miller, Roy and Straubing (Journal of Number Theory, 2006) for general MAJ~$circ mathrm{MOD}_m circ
{
m AND}_2$ circuits for any odd $m$. 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 $m
ot = 3$ quadratic
case as well as for
higher degrees.

@InProceedings{green_et_al:DagSemProc.07411.7,
  author =	{Green, Frederic and Roy, Amitabha},
  title =	{{Uniqueness of Optimal Mod 3 Circuits  for Parity}},
  booktitle =	{Algebraic Methods in Computational Complexity},
  pages =	{1--15},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2008},
  volume =	{7411},
  editor =	{Manindra Agrawal and Harry Buhrman and Lance Fortnow and Thomas Thierauf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2008/1305},
  URN =		{urn:nbn:de:0030-drops-13059},
  doi =		{10.4230/DagSemProc.07411.7},
  annote =	{Keywords: Circuit complexity, correlations, exponential sums}
}