On Relaxed Locally Decodable Codes for Hamming and Insertion-Deletion Errors

TL;DR

This paper investigates relaxed locally decodable codes (RLDCs) in Hamming and insertion-deletion error settings, establishing exponential lower bounds and demonstrating fundamental differences between these variants, with implications for DNA data storage technologies.

Contribution

It provides the first exponential lower bounds for 2-query binary RLDCs in Hamming errors and introduces novel variants of RLDCs in the insdel error setting, highlighting their differences.

Findings

Exponential lower bounds for 2-query Hamming RLDCs over binary alphabet.

A phase-transition behavior in codeword length for constant-query Hamming RLDCs.

Strict separation between Hamming RLDCs and Insdel RLDCs.

Abstract

Locally Decodable Codes (LDCs) are error-correcting codes $C:\Sigma^n\rightarrow \Sigma^m$ with super-fast decoding algorithms. They are important mathematical objects in many areas of theoretical computer science, yet the best constructions so far have codeword length $m$ that is super-polynomial in $n$, for codes with constant query complexity and constant alphabet size. In a very surprising result, Ben-Sasson et al. showed how to construct a relaxed version of LDCs (RLDCs) with constant query complexity and almost linear codeword length over the binary alphabet, and used them to obtain significantly-improved constructions of Probabilistically Checkable Proofs. In this work, we study RLDCs in the standard Hamming-error setting, and introduce their variants in the insertion and deletion (Insdel) error setting. Insdel LDCs were first studied by Ostrovsky and Paskin-Cherniavsky, and are further motivated by recent advances in DNA random access bio-technologies, in which the goal is to retrieve individual files from a DNA storage database. Our first result is an exponential lower bound on the length of Hamming RLDCs making 2 queries, over the binary alphabet. This answers a question explicitly raised by Gur and Lachish. Our result exhibits a "phase-transition"-type behavior on the codeword length for constant-query Hamming RLDCs. We further define two variants of RLDCs in the Insdel-error setting, a weak and a strong version. On the one hand, we construct weak Insdel RLDCs with with parameters matching those of the Hamming variants. On the other hand, we prove exponential lower bounds for strong Insdel RLDCs. These results demonstrate that, while these variants are equivalent in the Hamming setting, they are significantly different in the insdel setting. Our results also prove a strict separation between Hamming RLDCs and Insdel RLDCs.