Locally Decodable Codes from Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers

Abstract

A $k$-query locally decodable code (LDC) encodes an $n$-bit message $x$ as an $N$-bit codeword $C(x)$, such that one can probabilistically recover any bit $x_i$ of the message by querying only $k$ bits of the codeword $C(x)$, even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently vast improvements in upper bounds for the length of LDCs were achieved via constructions that rely on existence of certain special (“nice”) subsets of finite fields. In this work we extend the constructions of LDCs from “nice” subsets. We argue that further progress on upper bounds for LDCs via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number $m=2^t-1$ that has a prime factor $p>m^\gamma$ yields a family of $k(\gamma)$-query LDCs of length $\exp(n^{1/t})$. Conversely, if for some fixed $k$ and all $\epsilon>0$ one can use the “nice” subsets technique to obtain a family of $k$-query LDCs of length $\exp(n^\epsilon)$, then infinitely many Mersenne numbers have prime factors larger than currently known.