Crosscorrelation of Rudin–Shapiro-like polynomials

Abstract

We consider the class of Rudin-Shapiro-like polynomials, whose L 4 norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f (z) = f0 + f1z + · · · + f d z d is identified with the sequence (f0, f1, . . . , f d ) of its coefficients. From the L 4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. In this paper, we study the crosscorrelation properties of pairs of sequences associated to Rudin-Shapiro-like polynomials. We find an explicit formula for the crosscorrelation merit factor. A computer search is then used to find pairs of Rudin-Shapiro-like polynomials whose autocorrelation and crosscorrelation merit factors are simultaneously high. Pursley and Sarwate proved a bound that limits how good this combined autocorrelation and crosscorrelation performance can be. We find infinite families of polynomials whose performance approaches quite close to this fundamental limit.