Average hamming correlation for the cubic polynomial hopping sequences

Frequency hopping/time hopping (FH/TH) sequences have found wide applications in various modern FH/TH spread spectrum communications, namely ultra-wide bandwidth radio, multi-user radar, sonar systems, wireless sensor networks, and so on. The average Hamming correlation (AHC) is an important performance indicator of the FH/TH sequences. In order to evaluate the performance of the FH/TH sequences, it is important to find the tight theoretical bound on the AHC. The lower bound, which has not yet been previously reported, on the AHC for FH/TH sequences with respect to the size of the frequency slot set, the sequence length, the family size, the average Hamming autocorrelation and the average Hamming cross-correlation is established here. In addition, a new family of cubic and constant hopping sequences with large family size suitable for FH/TH code-division multiple-access systems is proposed. Further, the AHCs for the new cubic and constant hopping and some known cubic hopping sequences are derived analytically. It is shown that these cubic hopping sequences are optimal with respect to our new bound on the AHC.