Applications of additive sequence of permutations

Let X1X1 be the mm-vector (−r,−r+1,…,−1,0,1,…,r−1,r)(−r,−r+1,…,−1,0,1,…,r−1,r), m=2r+1m=2r+1, and X2,…,XnX2,…,Xn be permutations of X1X1. Then X1,X2,…,XnX1,X2,…,Xn is said to be an additive sequence of permutations (ASP) of order mm and length nn if the vector sum of every subsequence of consecutive permutations is again a permutation of X1X1. ASPs had been extensively studied and used to construct perfect difference families. In this paper, ASPs are used to construct perfect difference families and properly centered permutation matrices (which are related to radar arrays). More existence results on perfect difference families and properly centered permutation matrices are obtained.