Explicit uniquely decodable codes for space bounded channels that achieve list-decoding capacity

We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in one pass, and modifies at most a p fraction of the bits of the codeword.

• Explicit uniquely decodable codes for space bounded channels: Our main result is that for every 0 ≤ p < 1 4 , there exists a constant δ > 0 and a uniquely decodable code that is explicit (meaning that encoding and decoding are in poly-time) and has rate 1 -H(p) for channels with space n δ . This improves upon previous explicit codes by Guruswami and Smith, and Kopparty, Shaltiel and Silbak (FOCS 2019). Specifically, we obtain the same space and rate as earlier works, even though prior work gave only list-decodable codes (rather than uniquely decodable codes).

• Complete characterization of the capacity of space bounded channels: Together with a result by Guruswami and Smith showing the impossibility of unique decoding for p ≥ 1 4 , our techniques also give a complete characterization of the capacity R(p) of space n 1-o(1) channels, specifically: R(p) = 1 -H(p) 0 ≤ p < 1/4 0 p ≥ 1/4.

In particular, R(•) is not continuous at p = 1/4. This capacity is strictly larger than the capacity of Hamming channels for every 0 < p < 1 4 , and matches the capacity of list decoding, and binary symmetric channels in this range.