On the Sidel'nikov Sequences as Frequency-Hopping Sequences

Abstract

A (v, l, lambda)-FHS denotes a frequency-hopping sequence of length over a frequency set of size 1 with maximum out-of-phase Hamming autocorrelation lambda. Recently, Ding and Yin constructed two FHS families for a prime power q satisfying q = ef + 1 with positive integers e and f. Theorems 4 and 5 in their paper claim that these two FHS families include optimal (q - 1, e, f)-FHSs and (q - 1, e + 1, f - 1)-FHSs with respect to the Lempel-Greenberger bound, respectively. In this paper, we give counterexamples and make corrections to them. Furthermore, we observe that these FHSs are closely related to Sidel'nikov sequences. Based on our results on the spectrum of their Hamming autocorrelation values, we also correct the theorem on the spectrum of Hamming distances of nearly equidistant codes derived by Sidel'nikov.