Applications of additive sequence of permutations

Let X 1 be the m -vector ( − r , − r + 1 , … , − 1 , 0 , 1 , … , r − 1 , r ) , m = 2 r + 1 , and X 2 , … , X n be permutations of X 1 . Then X 1 , X 2 , … , X n is said to be an additive sequence of permutations (ASP) of order m and length n if the vector sum of every subsequence of consecutive permutations is again a permutation of X 1 . 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.