# A Construction of Optimal Frequency Hopping Sequence Set via Combination of Multiplicative and Additive Groups of Finite Fields

## TL;DR

This paper introduces a novel method combining multiplicative and additive groups of finite fields to construct optimal frequency hopping sequence sets that achieve the Peng-Fan bound, encompassing previous constructions.

## Contribution

It presents a new family of optimal FHS sets using both groups of finite fields, unifying and extending prior constructions with improved parameters.

## Key findings

- Constructs optimal FHS sets achieving Peng-Fan bound
- Includes all previous FHS sets based on finite fields
- Provides recursive methods for additional FHS set construction

## Abstract

In literatures, there are various constructions of frequency hopping sequence (FHS for short) sets with good Hamming correlations. Some papers employed only multiplicative groups of finite fields to construct FHS sets, while other papers implicitly used only additive groups of finite fields for construction of FHS sets. In this paper, we make use of both multiplicative and additive groups of finite fields simultaneously to present a construction of optimal FHS sets. The construction provides a new family of optimal $\left(q^m-1,\frac{q^{m-t}-1}{r},rq^t;\frac{q^{m-t}-1}{r}+1\right)$ frequency hopping sequence sets archiving the Peng-Fan bound. Thus, the FHS sets constructed in literatures using either multiplicative groups or additive groups of finite fields are all included in our family. In addition, some other FHS sets can be obtained via the well-known recursive constructions through one-coincidence sequence set.