Binary Sequences with Low Aperiodic Autocorrelations: Small Peak Sidelobe Levels

Binary sequences (binary codes), where the elements are −1 or +1, are useful in many fields, including communications, radar, sonar, mathematics, physics, and cryptography. This paper considers binary sequences with low aperiodic autocorrelations and focuses on the small peak sidelobe levels alongside the merit factor. Two families of binary sequences are considered, namely Rudin-Shapiro and Legendre sequences. For both families, we applied a heuristic algorithm to minimize the peak sidelobe levels for sequences of lengths up to 2^16 and 220−1, respectively. The main contribution of the article is two conjectures associated with Legendre sequences: (1) The obtained binary sequences with the best-known peak sidelobe levels have merit factor ≈5.0, (2) The number of elements that differ between the resulting binary sequences and the initial Legendre sequences follows a linear dependence on the sequence length (n), namely ≈0.01n. The Rudin-Shapiro sequences do not exhibit these properties, as worse peak sidelobe level and merit factor values were obtained. The number of elements that differ between the resulting binary sequences and the initial Rudin-Shapiro sequences is also much higher compared to that of the Legendre sequences.