On the Modulus in Matching Vector Codes

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 can be recovered by looking at $k$ $C(x)$, even if constant fraction has been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). modulus $m=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_r^{\alpha _r}$ may result in an MVC with $k\leq 2^r$ and $N=\exp (\exp (O((\log n)^{1-1/r} (\log \log n)^{1/r})))$. The $m$ is good it possible have $k<2^r$. numbers yield more efficient MVCs. Prior this work, there only finitely many numbers. All them were obtained via computer search form $m=p_1p_2$. In paper, we study _2}$. We show _2}$ good, then multiple $p_1^{\beta _1}p_2^{\beta must well. Given number _2}$, explicit method obtaining smaller same prime divisors. Our approach yields infinitely new