Research On Ideal Ternary Autocorrelation Binary Sequence Pairs Based On Generakized Cyclotomic Classes

Ideal sequences and sequence pairs with good autocorrelation and balance are widely used in engineering fields such as spread spectrum communication,data encryption,simulation systems,and positioning systems.However,due to technological limitations,there are few direct construction methods for sequences and sequence pairs.The number of sequences and sequence pairs obtained using computer exhaustive search and theoretical derivation methods is small and the performance is poor,which cannot meet the needs of practical applications.Therefore,scholars use some mathematical tools to assist in constructing sequences and sequence pairs,such as almost difference sets,difference sets,almost difference set pairs,etc.This thesis uses the generalized cyclotomic method to study the construction methods of almost difference set pairs,and obtains two new types of construction methods of almost difference set pairs through computer search algorithms.Firstly,through in-depth analysis of the classical cyclotomization algorithm and combining the operational ideas of Cartesian product and Chinese remainder theorem,a construction algorithm for the generalized cyclotomization class is proposed.On the basis of this construction algorithm,a nearly difference set pair search algorithm based on the generalized cyclotomic class is proposed,while achieving equivalent conversion between almost difference set pairs and binary sequence pairs in the algorithm.This thesis utilizes this search algorithm to obtain a large number of almost difference sets,providing data support for future research work.Secondly,the construction method of almost difference set pairs with a period length of pq is studied based on the generalized cyclotomic class of order 2-4.After analyzing,screening,and comparing the parameters,main peaks,and sub peaks of almost difference set pairs in search results,various new methods for constructing almost difference set pairs are proposed based on the different values of prime number p.Describe these construction methods in the form of theorems,use theoretical derivation to provide the proof process of the theorems,and verify the accuracy of the proposed theorems through examples.In addition,according to the equivalence relation between almost difference set pairs and ternary autocorrelation binary sequence pairs,a variety of new and balanced ideal ternary autocorrelation binary sequence pairs are obtained.Finally,the construction method of almost difference set pairs with a period length of5p is studied.Based on the generalized cyclotomy overG_5P,in the case of p≡3（mod4）,various new parameter forms of almost difference set pairs are proposed,and the autocorrelation function values of their corresponding feature sequence pairs are obtained based on the properties of almost difference set pairs.By analyzing the autocorrelation and balance of sequence pairs,we can construct balanced binary sequence pairs with good autocorrelation.