Computing the Ball Size of Frequency Permutations under Chebyshev Distance

Abstract

Let $S_n^{\lambda }$ be the set of all permutations over the multiset $\{\stackrel{\lambda }{\overbrace{1,...,1}},...,\stackrel{\lambda }{\overbrace{m,...,m}}\}$ where $n=m\lambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^{\lambda }$ 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 $O({\left(\genfrac{}{}{0ex}{}{2d\lambda }{d\lambda }\right)}^{2.376}\log n)$ time and $O({\left(\genfrac{}{}{0ex}{}{2d\lambda }{d\lambda }\right)}^2)$ space. The second one runs in $O((\genfrac{}{}{0ex}{}{2d\lambda }{d\lambda }\left)\right(\genfrac{}{}{0ex}{}{d\lambda +\lambda }{\lambda })\frac{n}{\lambda })$ time and $O((\genfrac{}{}{0ex}{}{2d\lambda }{d\lambda }))$ space. For small constants $\lambda$ and $d$, both are efficient in time and use constant storage space.

Submitted to ISIT 2011