关于循环差集的一个必要条件

On a Necessary Condition of Cyclic Difference Set

设v,k,λ都是正整数.一个(v,k,λ)-循环差集B={b_1,b_2,…,b_k}是k个不同的模v的剩余组成的集合,其中有对任何一个模v不同余0的数b,同余方程b_i-b_j≡b(mod v)都恰有λ组解,其中bi,bj都属于B.本文给出了(v,k,λ)-循环差集的一个必要条件,并计算出了阶为2和3的所有的循环差集.

Let v,k andλbe positive integers.Suppose that B={b 1,b 2…,b k}is a set of kresidues modulo v with the property that for any residue bthat is not congruential to 0(mod v)the congruence b i-b j≡b(mod v)has exactlyλsolution pairs(b i,b j)with b i and b j in B.Such a residue set is called a cyclic difference set.In this paper,we give a new necessary condition of cyclic difference set,and then compute all cyclic difference set of order 2 and 3.