Computing the ball size of frequency permutations under Chebyshev distance

Let be the set of all permutations over the multiset where n=mλ. A frequency permutation array (FPA) of minimum distance d is a subset of in which every two elements have distance at least d. FPAs have many applications related to error correcting codes. In coding theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived from the size of balls of certain radii.We propose two efficient algorithms that compute the ball size of frequency permutations under Chebyshev distance. Here it is equivalent to computing the permanent of a special type of matrix, which generalizes the Toepliz matrix in some sense. Both methods extend previous known results. The first one runs in time and space. The second one runs in time and space. For small constants λ and d, both are efficient in time and use constant storage space.