Oblivious Deletion Codes

TL;DR

This paper develops new deletion error-correcting codes in the oblivious model, bridging the gap between adversarial and random errors, with constructions that improve redundancy bounds and have applications in DNA storage.

Contribution

It introduces explicit, randomized, and list-decodable oblivious deletion codes, achieving redundancy bounds close to the theoretical limits and linking oblivious and adversarial deletion code constructions.

Findings

Constructed $t$ oblivious deletion codes with redundancy $ ilde{2t ext{log} n}$.

Explicit list-decodable codes yield oblivious deletion codes with similar parameters.

Randomized constructions achieve redundancy $ ilde{(t+1) ext{log} n}$ for oblivious deletions.

Abstract

We construct deletion error-correcting codes in the oblivious model, where errors are adversarial but oblivious to the encoder's randomness. Oblivious errors bridge the gap between the adversarial and random error models, and are motivated by applications like DNA storage, where the noise is caused by hard-to-model physical phenomena, but not by an adversary. (1) (Explicit oblivious) We construct $t$ oblivious deletion codes, with redundancy $\sim 2t\log n$, matching the existential bound for adversarial deletions. (2) (List decoding implies explicit oblivious) We show that explicit list-decodable codes yield explicit oblivious deletion codes with essentially the same parameters. By a work of Guruswami and H\r{a}stad (IEEE TIT, 2021), this gives 2 oblivious deletion codes with redundancy $\sim 3\log n$, beating the existential redundancy for 2 adversarial deletions. (3) (Randomized oblivious) We give a randomized construction of oblivious codes that, with probability at least $1-2^{-n}$, produces a code correcting $t$ oblivious deletions with redundancy $\sim(t+1)\log n$, beating the existential adversarial redundancy of $\sim 2t\log n$. (4) (Randomized adversarial) Studying the oblivious model can inform better constructions of adversarial codes. The same technique produces, with probability at least $1-2^{-n}$, a code correcting $t$ adversarial deletions with redundancy $\sim (2t+1)\log n$, nearly matching the existential redundancy of $\sim 2t\log n$. The common idea behind these results is to reduce the hash size by modding by a prime chosen (randomly) from a small subset, and including a small encoding of the prime in the hash.