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**Published in:** Dagstuhl Seminar Proceedings, Volume 7411, Algebraic Methods in Computational Complexity (2008)

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.

Frederic Green and Amitabha Roy. Uniqueness of Optimal Mod 3 Circuits for Parity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 7411, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@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/entities/document/10.4230/DagSemProc.07411.7}, URN = {urn:nbn:de:0030-drops-13059}, doi = {10.4230/DagSemProc.07411.7}, annote = {Keywords: Circuit complexity, correlations, exponential sums} }

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