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arXiv · 2408.14059

On the pseudorandomness of Parry--Bertrand automatic sequences

Pierre Popoli·Manon Stipulanti
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Abstract

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.

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Pierre Popoli, Manon Stipulanti. 2024-08-26. On the pseudorandomness of Parry--Bertrand automatic sequences. https://arxiv.org/abs/2408.14059

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