# On the Theoretical Gap of Channel Hopping Sequences with Maximum Rendezvous Diversity in the Multichannel Rendezvous Problem

## TL;DR

This paper investigates the theoretical limits of channel hopping sequences in multichannel rendezvous problems, proposing new sequences that improve the asymptotic approximation ratio and address open questions about optimality.

## Contribution

It introduces IDEAL-CH sequences with an asymptotic ratio of 2 and ORTHO-CH sequences for weaker rendezvous requirements, advancing the theoretical understanding of optimal channel hopping.

## Key findings

- IDEAL-CH achieves an asymptotic ratio of 2.
- ORTHO-CH sequences have period (2p+1)p, with p as the smallest prime ≥ N.
- The paper tightens the theoretical gap in rendezvous sequence design.

## Abstract

In the literature, there are several well-known periodic channel hopping (CH) sequences that can achieve maximum rendezvous diversity in a cognitive radio network (CRN). For a CRN with $N$ channels, it is known that the period of such a CH sequence is at least $N^2$. The asymptotic approximation ratio, defined as the ratio of the period of a CH sequence to the lower bound $N^2$ when $N \to \infty$, is still 2.5 for the best known CH sequence in the literature. An open question in the multichannel rendezvous problem is whether it is possible to construct a periodic CH sequence that has the asymptotic approximation ratio 1. In this paper, we tighten the theoretical gap by proposing CH sequences, called IDEAL-CH, that have the asymptotic approximation ratio 2. For a weaker requirement that only needs the two users to rendezvous on one commonly available channel in a period, we propose channel hopping sequences, called ORTHO-CH, with period $(2p +1)p$, where $p$ is the smallest prime not less than $N$.