Finding automatic sequences with few correlations

Vincent Jugé , Irène Marcovici

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.

Mots clés: Combinatorics on words; automata sequence; well-distributed sequence

Domaines: Mathématiques [math]

Vincent Jugé, Irène Marcovici. Finding automatic sequences with few correlations. RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), 2024, 58, pp.10. ⟨10.1051/ita/2024008⟩. ⟨hal-04522641⟩