Upper Bounds on Matching Families in $\mathbb{Z}_{pq}^n$

TL;DR

This paper establishes an upper bound on the size of matching families in the ring of integers modulo a product of two primes, advancing understanding relevant to locally decodable codes.

Contribution

It provides a new upper bound for matching families in rac{1}{2}( ext{pq})^{0.625n+0.125} in rac{ ext{pq}}{n} when p and q are primes with p/q approaching 1.

Findings

Upper bound of O((pq)^{0.625n+0.125}) for matching families.

Bound valid for fixed n, pa0b0a0b0a0b0 pa0b0a0b0a0b0 q, with p/qa0b0a0b0a0b0 1.

Improves previous bounds by Dvir et al.

Abstract

\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.}