# Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences

Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin-Shapiro sequences, and in general it produces what we call Golay-Rudin-Shapiro sequences. Calculations by Littlewood show that the Rudin-Shapiro sequences have low mean square autocorrelation. A sequence's peak sidelobe level is its largest magnitude of autocorrelation over all nonzero shifts. Høholdt, Jensen, and Justesen showed that there is some undetermined positive constant A such that the peak sidelobe level of a Rudin-Shapiro sequence of length 2^n is bounded above by A(1.842626…)^n, where 1.842626… is the positive real root of X^4-3 X-6. We show that the peak sidelobe level is bounded above by 5(1.658967…)^n-4, where 1.658967… is the real root of X^3+X^2-2 X-4. Any exponential bound with lower base will fail to be true for almost all n, and any bound with the same base but a lower constant prefactor will fail to be true for at least one n. We provide a similar bound on the peak crosscorrelation (largest magnitude of crosscorrelation over all shifts) between the sequences in each Rudin-Shapiro pair. The methods that we use generalize to all families of complementary pairs produced by the Golay-Rudin-Shapiro recursion, for which we obtain bounds on the peak sidelobe level and peak crosscorrelation with the same exponential growth rate as we obtain for the original Rudin-Shapiro sequences.

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