In Brute-Force Search of Correlation Bounds for Polynomials

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 previouslyanalyzed 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.