Tight bounds for stream decodable error-correcting codes

TL;DR

This paper explores the limits of stream decodable error-correcting codes for noisy channels, establishing bounds on code length and demonstrating the existence of near-quadratic and near-linear codes under computational constraints.

Contribution

It introduces new bounds for stream decodable codes and constructs near-quadratic and near-linear codes under bounded computational resources.

Findings

Existence of near-quadratic length stream decodable codes.

Impossibility of sub-quadratic length stream decodable codes.

Feasibility of near-linear length codes for linear function computation.

Abstract

In order to communicate a message over a noisy channel, a sender (Alice) uses an error-correcting code to encode her message $x$ into a codeword. The receiver (Bob) decodes it correctly whenever there is at most a small constant fraction of adversarial error in the transmitted codeword. This work investigates the setting where Bob is computationally bounded. Specifically, Bob receives the message as a stream and must process it and write $x$ in order to a write-only tape while using low (say polylogarithmic) space. We show three basic results about this setting, which are informally as follows: (1) There is a stream decodable code of near-quadratic length. (2) There is no stream decodable code of sub-quadratic length. (3) If Bob need only compute a private linear function of the input bits, instead of writing them all to the output tape, there is a stream decodable code of near-linear length.