# Crosscorrelation of Rudin-Shapiro-Like Polynomials

## TL;DR

This paper investigates the crosscorrelation properties of Rudin-Shapiro-like polynomials, deriving explicit formulas and identifying polynomial pairs with near-optimal combined autocorrelation and crosscorrelation merit factors.

## Contribution

It provides an explicit formula for the crosscorrelation merit factor and finds polynomial pairs with near-optimal combined correlation properties.

## Key findings

- Derived an explicit formula for crosscorrelation merit factor.
- Identified polynomial pairs with high autocorrelation and crosscorrelation merit factors.
- Found infinite families approaching the theoretical performance limit.

## 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.