差基的下界

The lower bound for difference bases

若集合A■Z满足A-A■{1,2,...,n},则A称为关于n的差基.差基应用于图的优美标记、集合的对称相交族和信号处理,与密码和编码理论有密切而深刻的联系.本文利用Fourier分析的方法,讨论一些参数的更高阶的Fourier系数的性质,进一步改进了差基的下界,从而改进了R′edei和Re′nyi(1949)、Leech(1956)以及Bernshteyn和Tait(2019)的结果.

A set A■Z is called a difference basis with respect to n if A-A■{1,2,...,n}.Finding the minimum size of a difference basis,while it is a natural question in combinatorial number theory in its own right,also has applications to graceful labelings of graphs,to symmetric intersecting families of sets,and to signal processing.It is closely and deeply related to the theory of cryptography and coding theory.Applying the method of Fourier analysis,we discuss the properties of higher order Fourier coefficients of some parameters,further improve the lower bound of the difference basis,and therefore improve the related results of Re′dei and Re′nyi(1949),Leech(1956),and Bernshteyn and Tait(2019).