# Three new classes of optimal frequency-hopping sequence sets

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## 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 (IEEE Trans Inf Theory 50(9):2149–2154, 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 \( \mathcal {C}_0=\{(\alpha ,\alpha ,\ldots ,\alpha )\in \mathbb {F}_q^n\,|\,\alpha \in \mathbb {F}_q\}. \) 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 et al. (IEEE Trans Inf Theory 55(7):3297–3304, 2009). 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.

## Keywords

Frequency-hopping sequence set Cyclic code Maximum distance separable (MDS) code Cyclotomic coset## Mathematics Subject Classification

94A55 94B05## Notes

### Acknowledgments

The authors are grateful to the three anonymous reviewers for their helpful comments and suggestions that improved an earlier version of this paper. The research of the first and third authors was partially supported by NTU Research Grant M4080456. The fourth author was supported by NSFC (Grant No. 11171370) and self-determined research funds of CCNU from the colleges’s basic research and operation of MOE (Grant No. CCNU14F01004). Part of this work was done when B. Chen was with Nanyang Technological University.

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