Chebyshev距离下频率置换球大小的计算
Computing the Ball Size of Frequency Permutations under Chebyshev Distance
摘要：
令$S_n^λ$是多集$\{\overbrace{1,...,1}^λ,...,\overbrace{m,...,m}^λ\}$上所有置换的集合，其中$n=mλ$。最小距离$d$的频率排列阵列（FPA）是$S_n^λ$的子集，其中每两个元素的距离至少为$d$。FPA具有许多与纠错码相关的应用。在编码理论中，Gilbert-Varshamov界和球填充界都是由一定半径的球的大小推导出来的。本文提出了两个计算Chebyshev距离下频率排列球大小的有效算法。这两种方法都推广了以前的已知结果。第一个在$O({2dλ\choose dλ}^{2.376}\log n)$时间和$O({2dλ\choose dλ}^{2})$空间中运行。第二个在$O({2dλ\choose dλ}{dλ+λ\choose λ}\frac{n}λ)$时间和$O({2dλ\choose dλ})$空间中运行。对于小常数$λ$和$d$，两者都在时间上是有效的，并且使用常数存储空间。
英文摘要：
Let $S_n^λ$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^λ,...,\overbrace{m,...,m}^λ\}$ where $n=mλ$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^λ$ 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 $O({2dλ\choose dλ}^{2.376}\log n)$ time and $O({2dλ\choose dλ}^{2})$ space. The second one runs in $O({2dλ\choose dλ}{dλ+λ\choose λ}\frac{n}λ)$ time and $O({2dλ\choose dλ})$ space. For small constants $λ$ and $d$, both are efficient in time and use constant storage space.
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