On the Modulus in Matching Vector Codes. (22nd September 2021) - Zhu, Lin; Li, Wen Ming; Zhang, Liang Feng
Abstract: A $k$ -query locally decodable code (LDC) $C$ allows one to encode any $n$ -symbol message $x$ as a codeword $C(x)$ of $N$ symbols such that each symbol of $x$ can be recovered by looking at $k$ symbols of $C(x)$, even if a constant fraction of $C(x)$ has been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). A modulus $m=p_1^{α _1}p_2^{α _2}… p_r^{α _r}$ may result in an MVC with $k≤ 2^r$ and $N=\exp (\exp (O((\log n)^{1-1/r} (\log \log n)^{1/r})))$ . The $m$ is good if it is possible to have $k<2^r$ . The good numbers yield more efficient MVCs. Prior to this work, there are only finitely many good numbers. All of them were obtained via computer search and have the form $m=p_1p_2$ . In this paper, we study good numbers of the form $m=p_1^{α _1}p_2^{α _2}$ . We show that if $m=p_1^{α _1}p_2^{α _2}$ is good, then any multiple of $m$ of the form $p_1^{β _1}p_2^{β _2}$ must be good as well. Given a good number $m=p_1^{α _1}p_2^{α _2}$, we show an explicit method of obtaining smaller good numbers that have the same prime divisors. Our approach yields infinitely many new good numbers.