On the Aperiodic Autocorrelations of Rotated Binary Sequences

Abstract

Binary sequences have found major applications in various technological domains such as radar and communication systems. A well-known computational problem is finding the lowest possible peak sidelobe level (PSL) among the set of a binary sequence $B$ and all binary sequences generated by rotations of $B$ . In this letter, we present some useful properties of rotated binary sequences, which allowed us to project the aforementioned problem to a perfectly balanced parallelizable algorithm. The proposed algorithm, altogether with its graphical processing unit (GPU) implementation, is significantly faster than the existing instruments. Thus, we were able to exhaust the search space of all m-sequences with lengths $2^{n}-1$ , for $18 \leq n \leq 20$ . Then, we reveal a complete list of all PSL-optimal Legendre sequences, with or without rotations, for lengths up to 432100 - out of computational reach until now. The numerical experiments suggest that the PSL value of all PSL-optimal Legendre sequences, with or without rotations, and with lengths $N$ greater than 235723, are strictly greater than $\sqrt {N}$ .