2017/02/24 by Katz, Daniel J., Lee, Sangman, Trunov, Stanislav A.
#11B83 #42A05 #94A55 #Complex Variables (math.CV) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1702.07697
We consider the class of Rudin-Shapiro-like polynomials, whose L4 norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z)=f0+f1 z + ⋯ + fd zd is identified with the sequence (f0,f1,…,fd) of its coefficients. From the L4 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.