# On the Modulus in Matching Vector Codes. (arXiv:2107.09830v1 [cs.IT])

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 $C(x)$ have been corrupted. Currently, the best known LDCs

are matching vector codes (MVCs). A modulus

$m=p_1^{alpha_1}p_2^{alpha_2}cdots p_r^{alpha_r}$ may result in an MVC with

$kleq 2^r$ and $N=exp(exp(O((log n)^{1-1/r} (loglog n)^{1/r})))$. The $m$

is {em good} if it is possible to have $k<2^r$. The good numbers yield more

efficient MVCs. Prior to this work, there are only {em 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

$m=p_1^{alpha_1}p_2^{alpha_2}$. We show that if

$m=p_1^{alpha_1}p_2^{alpha_2}$ is good, then any multiple of $m$ of the form

$p_1^{beta_1}p_2^{beta_2}$ must be good as well. Given a good number

$m=p_1^{alpha_1}p_2^{alpha_2}$, we show an explicit method of obtaining

smaller good numbers that have the same prime divisors. Our approach yields

{em infinitely many} new good numbers.