Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic

Authors:
raw_author_name: Xubin Hu
raw_affiliation_string: School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
raw_author_name: Lingfei Jin
raw_affiliation_string: Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai, China
raw_author_name: Liming Ma
raw_affiliation_string: School of Mathematical Sciences, University of Science and Technology of China, Hefei, China
raw_author_name: Chaoping Xing
raw_affiliation_string: School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, China

Abstract: Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths, where binary sequences attract great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p<sup>n</sup></i></sub> for a prime <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> and a positive integer <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> . In fact, all <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n</sup></i> + 1 rational places including the place at infinity of the rational function field over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p<sup>n</sup></i></sub> can form a cyclic structure under an automorphism of order <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n</sup></i> + 1. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n</sup></i> + 1 via cyclotomic function fields over Fpn for any odd prime <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> . Each family of binary sequences has size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n</sup></i> -2 and its correlation is upper bounded by 4+⎣2 · <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n/2</sup></i> ⎦. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p<sup>n</sup></i> +1 for odd prime <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> . Moreover, our sequences can be constructed explicitly and have competitive parameters.