On the pseudorandomness of Parry-Bertrand automatic sequences

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.