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.

翻译：在通信和遥感中,使用周期性自动关系和交叉关系序列。 Golay 和 Shapiro 独立设计了一个循环构造, 产生双胞胎双胞胎。 在最简单的例子中, 建筑产生Rudin- Shapiro序列, 一般来说, 它产生我们称之为 Golay- Rudin- Shapiro 序列。 Littlewood 的计算显示, Rudin- Shapiro 序列的平面自动变异为平面变异。 一个序列的峰顶边线水平是它在所有非零位变异中最大的自动变异规模。 H~ holdt, Jensen, 和 Justesensen 显示, 有一些不确定的正数不变不变值 $A 。 因此, Rudin- Shapiro 序列的顶端水平是 $ 2⁄ 美元, 平面平面平面平面平面平面平面平面平方平面平面平面平面平面平面平面平面平面平面平方平面平面平面平面平面平面平面平面平面平面平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平