We report on some initial results of a brute-force search for determining the maximum correlation between degree-$d$ polynomials modulo $p$ and the $n$-bit mod $q$ function. For various settings of the parameters $n,d,p,$ and $q$, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previously-analyzed settings of parameters, where non-symmetric polynomials yield the maximum correlation.

We also prove new properties of maximum-correlation polynomials, and use those to obtain a new setting of parameters where those polynomials are not symmetric.