Frequency-Hopping Sequence Sets With Low Average and Maximum Hamming Correlation

TL;DR

This paper investigates the average Hamming correlation in frequency-hopping sequences, providing new constructions with optimal average correlation and analyzing their properties for improved FHMA system performance.

Contribution

It introduces novel constructions of FHS sets with optimal average Hamming correlation using cyclotomy and interleaving methods, and analyzes their correlation properties.

Findings

Calculated AHC of known FHS sets with optimal MHC.

Proved uniformly distributed FHS sets have optimal AHC.

Presented cyclotomy-based constructions with optimal AHC.

Abstract

In frequency-hopping multiple-access (FHMA) systems, the average Hamming correlation (AHC) among frequency-hopping sequences (FHSs) as well as the maximum Hamming correlation (MHC) is an important performance measure. Therefore, it is a challenging problem to design FHS sets with good AHC and MHC properties for application. In this paper, we analyze the AHC properties of an FHS set, and present new constructions for FHS sets with optimal AHC. We first calculate the AHC of some known FHS sets with optimal MHC, and check their optimalities. We then prove that any uniformly distributed FHS set has optimal AHC. We also present two constructions of FHS sets with optimal AHC based on cyclotomy. Finally, we show that if an FHS set is obtained from another FHS set with optimal AHC by an interleaving, it has optimal AHC.