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.