| Abstract: |
| The Rudin-Shapiro polynomials are a sequence of polynomials with coefficients in $\{+1,-1\}$ arising from a recursive construction that gives them properties of interest in combinatorics, harmonic analysis, dynamical systems, and information theory. When we identify a polynomial with the sequence of its coefficients, a typical Rudin-Shapiro polynomial has very low autocorrelation: the mean square value of inner products of the sequence with translates of itself is considerably smaller than that for a typical random binary sequence. Low autocorrelation makes sequences useful for synchronization in communications networks. To avoid confusing one user`s sequence with another`s, we demand pairs of sequences that also have low crosscorrelation, that is, small mean square value of inner products between one sequence and all translates of the other. We investigate pairs of Rudin-Shapiro-Like polynomials, a generalization of Shapiro`s original recursive construction, and derive a general formula for their mean square crosscorrelation. Borwein and Mossinghoff had shown that there are Rudin-Shapiro-like polynomials with low autocorrelation, and now we have found pairs of polynomials with low autocorrelation and low mutual crosscorrelation. There are bounds on how small these quantities can be, and we have found infinite families that achieve the bounds. |
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