Frequency-Hopping Sequences With Optimal Average Hamming Correlation and Their Applications in Energy and Spectrum Harvesting Technologies Area

Mingyue Fan

doi:10.1109/access.2020.3045611

Abstract

The research of cyclotomy theory can be traced to Gauss and it has been applied to many fields such as cryptography, coding theory, and combinatorics. According to $v$ prime numbers or compound words, the incision on the residue-like ring ${\mathbb Z}_{v}$ can be separated to classic incision or general incision. In this work, a kind of extended generalized cyclotomic classes is introduced. Based on this tangent method, a class of frequency hopping sequence set with the best average Hamming correlation is proposed.

Highlights

Suppose Zv is an integer ring modulo v
Suppose that Z∗v has a multiplicative sub-group D0, and h1, . . . , hd−1 is the elements of Z∗v so that for all i, Di = hiD0 and the cosets Di can be called as generalized cyclotomic classes when v is composite, and classical cyclotomic classes when v is prime
Proof: The length, family size, and alphabet size of X follows directly from its definition. We prove that it has optimal average Hamming correlation

Summary

INTRODUCTION

Suppose Zv is an integer ring modulo v. Hd−1 is the elements of Z∗v so that for all i, Di = hiD0 and the cosets Di can be called as generalized cyclotomic classes when v is composite, and classical cyclotomic classes when v is prime. The first objective of this work is to study an approach of extended generalized cyclotomic classes of order n and their basic properties and their cyclotomic numbers. This generalizes the study of order two’ cyclotomic classes. Another objective is using order n’s extended generalized cyclotomic classes to obtain highly Hamming correlational frequencyhopping sequence sets.

GENERALIZED CYCLOTOMY

CONCLUSION