New classes of generalized zero-difference balanced functions and their applications

ABSTRACT

Generalized zero-difference balanced (G-ZDB) function is a generalization of zero-difference balanced (ZDB) function, which has many important applications in coding and cryptography. In this paper, three new classes of G-ZDB functions are presented based on the generalized cyclotomic classes. Our constructions give new parameters of G-ZDB functions not covered in the literature. As applications, frequency-hopping sequences with optimal Hamming autocorrelation are obtained by using the newly constructed G-ZDB functions.

KEYWORDS:

• Generalized zero-difference balanced (G-ZDB) functions
• generalized cyclotomic classes
• frequency-hopping sequences

Nomenclature

$\mathit{A}\mathbf{,}\mathit{B}$ set

$f$ function

$S$ set

$N$ set of natural number

$\delta$ size of a set

$n,\lambda$ integers

$g$ primitive root

$\phi (\cdot )$ Euler function

${\mathit{D}}_i^{(n)}$ generalized cyclotomic classes

$\mathrm{\varnothing }$ empty set

$Z_n$ set of integers modulo $n$

$Z_n^{\ast }$ set of invertible elements in $Z_n$

$\eta (\cdot )$ bijection function

$S_1^{(j,i)}$, $T_1^{(j,i)}$ sets

${\mathrm{\Delta }}_1(a)$, ${\mathrm{\Delta }}_2(a)$ size of a set related to $a$

$F$ set of frequency slot

${\mathit{x}}_i$ frequency sequence

$X$ set of frequency sequences

$H_{x_i,x_j}(\tau )$ periodic Hamming correlation function

$H({\mathit{x}}_i)$ maximum periodic Hamming autocorrelation

Acknowledgement

The authors would like to thank the reviewers and editors for their detailed and constructive comments, which substantially improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (No. 61772292, 61772476), Natural Science Foundation of Fujian Province (No. 2015J01237), Fujian Normal University Innovative Research Team (No. IRTL1207).