Finding automatic sequences with few correlations

Although automatic sequences are algorithmically very simple, some of them have pseudorandom properties. In particular, some automatic sequences such as the Golay–Shapiro sequence are known to be 2-uncorrelated, meaning that they have the same correlations of order 2 as a uniform random sequence. Ho...

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Bibliographic Details
Published inRAIRO. Informatique théorique et applications Vol. 58; p. 10
Main Authors Jugé, Vincent, Marcovici, Irène
Format Journal Article
LanguageEnglish
Published EDP Sciences 2024
Subjects
Mathematics
well-distributed sequence
Combinatorics on words
automata sequence
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Summary:Although automatic sequences are algorithmically very simple, some of them have pseudorandom properties. In particular, some automatic sequences such as the Golay–Shapiro sequence are known to be 2-uncorrelated, meaning that they have the same correlations of order 2 as a uniform random sequence. However, the existence of ℓ-uncorrelated automatic sequences (for ℓ ⩾ 3) was left as an open question in a recent paper of Marcovici, Stoll and Tahay. We exhibit binary block-additive sequences that are 3-uncorrelated and, with the help of analytical results supplemented by an exhaustive search, we present a complete picture of the correlation properties of binary block-additive sequences of rank r ⩽ 5, and ternary sequences of rank r ⩽ 3.
ISSN:0988-3754
2804-7346
1290-385X
DOI:10.1051/ita/2024008