$\mathbb{Z}_{pq}^n$中匹配族的上界
Upper Bounds on Matching Families in $\mathbb{Z}_{pq}^n$
摘要：
\textit{Matching families}是构造{ \em局部可解码码}（LDCs）的主要成分之一，并且具有恒定数量查询的LDCs的最佳已知构造是基于匹配族。确定$\mathbb{Z}_m^n$中任意匹配族的最大规模是一个有趣的问题，其中$\mathbb{Z}_m$是模$m$的整数环.本文给出了$\mathbb{Z}_{pq}^n$中任意匹配族的大小的$O((pq)^{0.625n+0.125})$的一个上界，其中$p$和$q$是两个不同的素数.当$n$、$p\rightarrow \infty$和$p/q\rightarrow 1$为常数时，我们的界是有效的。我们的结果提高了Dvir的上界{ \it et al.}
英文摘要：
\textit{Matching families} are one of the major ingredients in the construction of {\em locally decodable codes} (LDCs) and the best known constructions of LDCs with a constant number of queries are based on matching families. The determination of the largest size of any matching family in $\mathbb{Z}_m^n$, where $\mathbb{Z}_m$ is the ring of integers modulo $m$, is an interesting problem. In this paper, we show an upper bound of $O((pq)^{0.625n+0.125})$ for the size of any matching family in $\mathbb{Z}_{pq}^n$, where $p$ and $q$ are two distinct primes. Our bound is valid when $n$ is a constant, $p\rightarrow \infty$ and $p/q\rightarrow 1$. Our result improves an upper bound of Dvir {\it et al.}
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