Year: 2013
Type: preprint
Abstract: We prove new upper bounds on the size of families of vectors in $\Z_m^n$ with restricted modular inner products, when $m$ is a large integer. More formally, if $\vec{u}_1,\ldots,\vec{u}_t \in \Z_m^n$ and $\vec{v}_1,\ldots,\vec{v}_t \in \Z_m^n$ satisfy $\langle\vec{u}_i,\vec{v}_i\rangle\equiv0\pmod m$ and $\langle\vec{u}_i,\vec{v}_j\rangle\not\equiv0\pmod m$ for all $i\neq j\in[t]$, we prove that $t \leq O(m^{n/2+8.47})$. This improves a recent bound of $t \leq m^{n/2 + O(\log(m))}$ by \cite{BDL13} and is the best possible up to the constant 8.47 when $m$ is sufficiently larger than $n$. The maximal size of such families, called `Matching-Vector families', shows up in recent constructions of locally decodable error correcting codes (LDCs) and determines the rate of the code. Using our result we are able to show that these codes, called Matching-Vector codes, must have encoding length at least $K^{19/18}$ for $K$-bit messages, regardless of their query complexity. This improves a known super linear bound of $ K2^{Ω({\sqrt{\log K}})}$ proved in \cite{DGY11}.
Source: arXiv (Cornell University)
Authors: Dvir, Zeev, Hu, Guangda
Institution: Princeton University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2402234829