New Constructions for Near-Optimal Sets of Frequency-Hopping Sequences via the Gaussian Periods in Finite Fields

Abstract

Frequency-hopping sequences (FHSs) have been widely applied in frequency-hopping code-division multiple-access (FH-CDMA) systems, since they can be used for transmitting messages efficiently along with switching frequencies at set intervals by each sender. The performance of the FHSs has a great impact on the performance of FH-CDMA systems. The optimality achieving exactly the Peng-Fan bounds is an important performance measure. However, optimal sets of FHSs do not always exist for all lengths and alphabet sizes. Thus, it is meaningful to seek and design more near-optimal FHS sets whose parameters are near to achieving the Peng-Fan bounds. Let $q$ be a power of a prime. In this paper, we present some classes of near-optimal sets of FHSs, whose parameters are $\left({\frac {3(q+1)}{2},\frac {2(q-1)}{3},3;q}\right)$ with $q\equiv 1 \pmod {12}$ , $\left({\frac {q+1}{k},{k(q-1)},2;q}\right)$ with even $\frac {q+1}{k}$ , $\left({2(q^{2}+1),\frac {q^{2}-1}{2},2(q+1);q}\right)$ with $q \equiv 3 \pmod {4}$ , $(19,18,4;7)$ , $(7,9,3;4)$ and $(91,45,7;16)$ respectively. Most importantly, these classes of near-optimal sets of FHSs have new parameters which are not covered in the foregoing literature.

Keywords

• Frequency-hopping sequence
• Peng-Fan bound
• cyclotomic class
• Gaussian period
• finite fields