# On the Hamming Auto- and Cross-correlation Functions of a Class of Frequency Hopping Sequences of Length $ p^{n} $

## TL;DR

This paper constructs a new class of frequency hopping sequences of length p^n using Ding-Helleseth cyclotomic classes, analyzing their correlation properties and demonstrating their optimality in average Hamming correlation.

## Contribution

It introduces a novel class of FHSs based on generalized cyclotomic classes and investigates their correlation functions, showing their optimality.

## Key findings

- The constructed FHSs have optimal average Hamming correlation.
- Correlation functions are explicitly characterized for the sequences.
- The sequences are particularly analyzed for p ≡ 3 mod 4.

## Abstract

In this paper, a new class of frequency hopping sequences (FHSs) of length $ p^{n} $ is constructed by using Ding-Helleseth generalized cyclotomic classes of order 2, of which the Hamming auto- and cross-correlation functions are investigated (for the Hamming cross-correlation, only the case $ p\equiv 3\pmod 4 $ is considered). It is shown that the set of the constructed FHSs is optimal with respect to the average Hamming correlation functions.