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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| Published in | RAIRO. Informatique théorique et applications Vol. 58; p. 10 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
EDP Sciences
2024
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| Subjects | |
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| Abstract | 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. |
|---|---|
| AbstractList | 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. 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. |
| Author | Jugé, Vincent Marcovici, Irène |
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| Cites_doi | 10.1017/CBO9780511546563 10.1515/udt-2016-0021 10.4064/aa140-4-5 10.5802/aif.1089 10.4064/aa135-4-1 10.2478/udt-2020-0001 10.4153/CJM-2017-053-1 |
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| Keywords | well-distributed sequence Combinatorics on words automata sequence |
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| Title | Finding automatic sequences with few correlations |
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