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 fanin $2$. We
also prove that the suboptimal circuits exhibit a stepped behavior:
any suboptimal 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.
BibTeX  Entry
@InProceedings{green_et_al:DSP:2008:1305,
author = {Frederic Green and Amitabha Roy},
title = {Uniqueness of Optimal Mod 3 Circuits for Parity},
booktitle = {Algebraic Methods in Computational Complexity},
year = {2008},
editor = {Manindra Agrawal and Harry Buhrman and Lance Fortnow and Thomas Thierauf},
number = {07411},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Internationales Begegnungs und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1305},
annote = {Keywords: Circuit complexity, correlations, exponential sums}
}
Keywords: 

Circuit complexity, correlations, exponential sums 
Collection: 

07411  Algebraic Methods in Computational Complexity 
Issue Date: 

2008 
Date of publication: 

15.01.2008 