3-受限匹配向量族的改进次上指数上界
Improved Subexponential Upper Bounds for $3$-Restricted Matching Vector Families
AI总结：
该研究针对$m \boldsymbol{\text{≤}} \boldsymbol{\text{√}}n$的情况，改进了3-受限匹配向量族的大小上界，其证明基于新的多项式方法论证，可用于构建更优的局部可解码码。
AI中文摘要：
匹配向量族（MVFs）由$\boldsymbol{\text{Z}}_m^n$中的两个有序向量列表定义，其内积满足模整数$m$的特定余数模式。最著名的是，受限MVFs用于构建已知最优的常查询局部可解码码（LDCs）。我们证明了当$m \boldsymbol{\text{≤}} \boldsymbol{\text{√}}n$时，$\boldsymbol{\text{Z}}_m^n$中3-受限MVFs的大小上界为$\boldsymbol{2^{O(\boldsymbol{\text{√}}n \boldsymbol{\text{log}}n \boldsymbol{\text{log}}m)}}$，大幅改进了Bhowmick、Dvir和Lovett（STOC'13、SICOMP'14）之前的最优界$\boldsymbol{2^{O(n/\boldsymbol{\text{log}}n)}}$。我们的证明依赖于一种新的多项式方法论证，用于控制匹配向量和集的碰撞。
英文摘要：
Matching Vector families (MVFs) are defined by two ordered lists of vectors in $\mathbb{Z}_m^n$ whose inner products satisfy specific residue patterns modulo an integer $m$. Most famously, restricted MVFs are used to construct the best-known constant-query Locally Decodable codes (LDCs). We prove an upper bound of $2^{O\left(\sqrt{n\log n \log m}\right)}$ on the size of $3$-restricted MVFs in $\mathbb{Z}_m^n$ for $m \leq \sqrt{n}$, substantially improving on the previous best bound of $2^{O(n/\log n)}$ by Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14). Our proof relies on a new polynomial method argument that controls collisions in sumsets of matching vectors.
