Computing the ball size of frequency permutations under Chebyshev distance

Abstract

Let Snλ be the set of all permutations over the multiset {1,…,1︷λts,m,…,m︷λ} where n=mλ. A frequency permutation array (FPA) of minimum distance d is a subset of Snλ 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 O2dλdλ2.376logn time and O2dλdλ2 space. The second one runs in O2dλdλdλ+λλnλ time and O2dλdλ space. For small constants λ and d, both are efficient in time and use constant storage space.