The correlation measure is a testimony of the pseudorandomness of a sequence $\infw{s}$ and provides information about the independence of some parts of $\infw{s}$ and their shifts. Combined with the well-distribution measure, a sequence possesses good pseudorandomness properties if both measures are relatively small. In combinatorics on words, the famous $b$-automatic sequences are quite far from being pseudorandom, as they have small factor complexity on the one hand and large well-distribution and correlation measures on the other. This paper investigates the pseudorandomness of a specific family of morphic sequences, including classical $b$-automatic sequences. In particular, we show that such sequences have large even-order correlation measures; hence, they are not pseudorandom. We also show that even- and odd-order correlation measures behave differently when considering some simple morphic sequences.

翻译：相关度量是序列$\infw{s}$伪随机性的一个表征，它提供了关于$\infw{s}$的某些部分及其移位之间独立性的信息。结合均匀分布度量，若一个序列的这两个度量值都相对较小，则该序列具有良好的伪随机性。在词组合学中，著名的$b$-自动序列与伪随机性相去甚远，因为它们一方面具有较小的因子复杂度，另一方面却具有较大的均匀分布度量和相关度量。本文研究了一类特定形态序列（包括经典$b$-自动序列）的伪随机性。特别地，我们证明此类序列具有较大的偶数阶相关度量，因此它们不是伪随机的。我们还表明，在考虑某些简单形态序列时，偶数阶与奇数阶相关度量的表现存在差异。