Binary Sequences With a Low Correlation via Cyclotomic Function Fields

Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field
$\mathbb {F}_{2^{n}}$
for a positive integer
$n$
. It is often overlooked in this community that all
$2^{n}+1$
rational places (including “the place at infinity”) of the rational function field over
$\mathbb {F}_{2^{n}}$
form a cyclic structure under an automorphism of order
$2^{n}+1$
. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length
$2^{n}+1$
via cyclotomic function fields over
$\mathbb {F}_{2^{n}}$
. Each family of our sequences has size
$2^{n}-1$
and its correlation is upper bounded by
$\lfloor 2^{(n+2)/2}\rfloor $
. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length
$2^{n}+1$
. Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length
$2^{n}-1$
for even
$n$
, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.