Computing the Ball Size of Frequency Permutations under Chebyshev Distance

Let S^λ_n be the set of all permutations over the multiset {1,⋯,1,λ ⋯,m,⋯,m λ} where n = mλ. A frequency permutation array (FPA) of minimum distance d is a subset of S^λ_n 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. Both methods extend previous known results. The first one runs in O ((2dλ dλ)^2.376log n) time and O ((2dλ dλ)²) space. The second one runs in O ((2dλ dλ) (dλ+λ λ)n/λ) time and O ((2dλ dλ)) space. For small constants λ and d, both are efficient in time and use constant storage space.