Optimal Frequency Hopping Sequences: Auto-and Cross-Correlation Properties

To begin with, we briefly present the application of the frequency hopping sequence in radar and communication systems (such as bluetooth), and comprehensively introduce some useful methods and important meaning of recent researches on frequency hopping sequence. Meanwhile, we give a concrete mathematical model for the frequency hopping sequence problem and define two different optimalities. After retrospecting some relevant basic conceptions and important theorems, meanwhile giving a concrete description and comparison with two different bounds, we expand a series of in-depth researches and constructions on the optimal frequency hopping sequence. We mainly use the tools of combinatorial design theory and algebra.1. In methods of combinatorial design theory, we briefly introduce the combinatorial research development and outcomes up to date, many of which are for a single sequence. After that, we use the set-theoretic perspective to change the single frequency hopping sequence problem into a combinatorial model called partition type difference packing and give the corresponding parameters. Then we focus our attentions on the constructions of optimal single sequence with v=3m-1, m even. In this situation, the partition type difference packing corresponding to the optimal sequence can be obtained from cyclic (3, 2)-frame(2^3t+1). In the following, we give a direct construction of the cyclic (3, 2)-frame(2^p) with prime p≡1 (mod 6). Then, we define a new combinatorial model called double difference matrix to construct the cyclic (3,2)-frame((6l+2)^p) with prime p≡7, 13 (mod 18). By the recursive construction of double difference matrix combined with difference matrix, we get a series of double difference matrices. Then, we can construct all the cyclic (3,2)-frame((24l+20)^p) with prime p≡7,13 (mod 18). We also put forward a new combinatorial model called balanced nested difference packing with t spanning partition type sub difference packings which is equivalent to FH sequences family and give the corresponding parameters.2. In methods of algebra, we first use the properties of cyclotomic class and cyclotomic number to construct balanced nested difference packing with t spanning partition type sub difference packings and prove the optimality or almost optimality of the corresponding frequency hopping sequences family. We also use trace function to construct frequency hopping sequences family. By the properties of trace function, character function and Gaussian sum, we can verify that the corresponding frequency hopping sequences family is optimal when the represent elements belong to the different cyclotomic classes. Finally, we define a new algebraic function satisfying certain properties and compound it with trace function to construct frequency hopping sequences family. By the properties of the newly defined algebraic, trace function and character function, we can also verify that the corresponding frequency hopping sequences family is optimal. Meanwhile, we give some concrete algebraic functions satisfying the desired properties.