Three new classes of optimal frequency-hopping sequence sets

TL;DR

This paper introduces three new classes of optimal frequency-hopping sequence sets constructed via cyclic codes, addressing open questions and demonstrating their optimality with respect to known bounds.

Contribution

It provides necessary and sufficient conditions for codeword equivalence classes in cyclic codes, leading to the construction of new optimal FHS sets and clarifying the relationship between key bounds.

Findings

Constructed three new classes of optimal FHS sets.

Established conditions for codeword equivalence class sizes.

Showed Peng-Fan bounds are identical.

Abstract

The study of frequency-hopping sequences (FHSs) has been focused on the establishment of theoretical bounds for the parameters of FHSs as well as on the construction of optimal FHSs with respect to the bounds. Peng and Fan (2004) derived two lower bounds on the maximum nontrivial Hamming correlation of an FHS set, which is an important indicator in measuring the performance of an FHS set employed in practice. In this paper, we obtain two main results. We study the construction of new optimal frequency-hopping sequence sets by using cyclic codes over finite fields. Let $\mathcal{C}$ be a cyclic code of length $n$ over a finite field $\mathbb{F}_q$ such that $\mathcal{C}$ contains the one-dimensional subcode $ \mathcal{C}_0=\{(\alpha,\alpha,\cdots,\alpha)\in \mathbb{F}_q^n\,|\,\alpha\in \mathbb{F}_q\}. $ Two codewords of $\mathcal{C}$ are said to be equivalent if one can be obtained from the other through applying the cyclic shift a certain number of times. We present a necessary and sufficient condition under which the equivalence class of any codeword in $\mathcal{C}\setminus\mathcal{C}_0$ has size $n$. This result addresses an open question raised by Ding {\it et al.} in \cite{Ding09}. As a consequence, three new classes of optimal FHS sets with respect to the Singleton bound are obtained, some of which are also optimal with respect to the Peng-Fan bound at the same time. We also show that the two Peng-Fan bounds are, in fact, identical.