Computing the Ball Size of Frequency Permutations under Chebyshev Distance

TL;DR

This paper introduces two efficient algorithms to compute the size of balls in frequency permutation spaces under Chebyshev distance, aiding error correction code design.

Contribution

The paper extends existing methods by providing two novel algorithms for calculating ball sizes in frequency permutation spaces under Chebyshev distance.

Findings

First algorithm runs in $O({2d ext{lambda} race d ext{lambda}}^{2.376} ext{log} n)$ time.

Second algorithm runs in $O({2d ext{lambda} race d ext{lambda}} {d ext{lambda}+ ext{lambda} race ext{lambda}} rac{n}{ ext{lambda}})$ time.

Both algorithms are efficient for small constants $ ext{lambda}$ and $d$, with low storage requirements.

Abstract

Let $S_n^\lambda$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^{\lambda},...,\overbrace{m,...,m}^\lambda\}$ 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 $O({2d\lambda \choose d\lambda}^{2.376}\log n)$ time and $O({2d\lambda \choose d\lambda}^{2})$ space. The second one runs in $O({2d\lambda \choose d\lambda}{d\lambda+\lambda\choose \lambda}\frac{n}{\lambda})$ time and $O({2d\lambda \choose d\lambda})$ space. For small constants $\lambda$ and $d$, both are efficient in time and use constant storage space.