Amortized Locally Decodable Codes for Insertions and Deletions

TL;DR

This paper introduces a method to convert Hamming locally decodable codes into insertion/deletion resilient codes with constant rate, locality, and error tolerance, advancing error correction in resource-limited settings.

Contribution

It provides a Hamming-to-Insdel compiler that preserves key properties and constructs an ideal amortized Hamming LDC satisfying specific conditions in relaxed settings.

Findings

Achieves constant amortized locality, rate, and error tolerance for Insdel aLDCs.

Provides a Hamming-to-Insdel compiler with minimal locality blow-up.

Constructs an ideal amortized Hamming LDC in resource-bounded settings.

Abstract

Locally Decodable Codes (LDCs) are error correcting codes which permit the recovery of any single message symbol with a low number of queries to the codeword (the locality). Traditional LDC tradeoffs between the rate, locality, and error tolerance are undesirable even in relaxed settings where the encoder/decoder share randomness or where the channel is resource-bounded. Recent work by Blocki and Zhang initiated the study of Hamming amortized Locally Decodable Codes (aLDCs), which allow the local decoder to amortize their number of queries over the recovery of a small subset of message symbols. Surprisingly, Blocki and Zhang construct asymptotically ideal (constant rate, constant amortized locality, and constant error tolerance) Hamming aLDCs in private-key and resource-bounded settings. While this result overcame previous barriers and impossibility results for Hamming LDCs, it is not clear whether the techniques extend to Insdel LDCs. Constructing Insdel LDCs which are resilient to insertion and/or deletion errors is known to be even more challenging. Our first contribution is to provide a Hamming-to-Insdel compiler which transforms any amortized Hamming LDC that satisfies a particular property to amortized Insdel LDC while asymptotically preserving the rate, error tolerance and amortized locality. Prior Hamming-to-Insdel compilers worked for arbitrary Hamming LDCs, but incurred an undesirable polylogarithmic blow-up in the locality. Our second contribution is a construction of an ideal amortized Hamming LDC which satisfies our special property in the relaxed settings where the sender/receiver share randomness or where the channel is resource bounded. Taken together, we obtain ideal Insdel aLDCs in private-key and resource-bounded settings with constant amortized locality, constant rate and constant error tolerance.