  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. H{\o}holdt,
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 $A(1.842626\ldots)^n$, where $1.842626\ldots$ is the
positive real root of $X^4-3 X-6$. We show that the peak sidelobe level is
bounded above by $5(1.658967\ldots)^{n-4}$, where $1.658967\ldots$ 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.
