Binary sequences with low correlation via cyclotomic function fields

TL;DR

This paper introduces a new explicit method for constructing binary sequences with low correlation using cyclotomic function fields, offering longer sequences with smaller correlation than some existing sequences.

Contribution

It leverages the cyclic structure of rational places in function fields to construct binary sequences with improved length and correlation properties.

Findings

Sequences have length $2^n+1$ with correlation bounded by $loor{2^{(n+2)/2}}$

Sequence families have size $2^n-1$

Sequences outperform Gold sequences in length and correlation for even n

Abstract

Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the multiplicative cyclic group of $\F_{2^n}$. It is often overlooked in this community that all $2^n+1$ rational places (including "place at infinity") of the rational function field over $\F_{2^n}$ 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 families of binary sequences of length $2^n+1$ via the finite field $\F_{2^n}$. Each family of sequences has size $2^n-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor$. Our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^n-1$ for even $n$, we have larger length and smaller correlation although the family size of our sequences is slightly smaller.