Title: 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
$f(z)=f_0+f_1 z + \cdots + f_d z^d$ is identified with the sequence
$(f_0,f_1,\ldots,f_d)$ 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.