Tight bounds for stream decodable error-correcting codes

TL;DR

This work introduces the concept of stream-decodable error-correcting codes, where a space-bounded, single-pass decoder must recover the original message from a corrupted codeword stream. It establishes a near-quadratic encoding length benchmark, proving a corresponding upper bound via a construction that repeats locally decodable codes with soft information and list-decoding techniques, and a matching information-theoretic lower bound showing $m(n)= ilde{oldsymbol{ obreak ext{-}}}rac{n^2}{s(n)}$ is necessary for constant success. It also shows a contrasting result for decoding linear functions: with near-linear encoding length, there exist schemes that compute private linear functions of the input in space $ ilde{O}(s(n))$, achieved by tensoring locally decodable codes and employing a recursive, low-space decoding strategy. The paper carefully engineers parameter regimes to balance error resilience up to $1/4- obreak\varepsilon$, encoding length, and decoding space, and it positions these stream-coding results in relation to GuptaZ23 and locally decodable codes. Overall, it advances the understanding of how streaming, space-bounded decoders interact with adversarial channels and what tradeoffs arise between rate, resilience, and computability of outputs. All mathematical expressions are presented with explicit $...$ delimiters to ensure precise formalism.

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.

Theorems & Definitions (51)

• Theorem 1.0
• Theorem 1.0
• Theorem 1.0
• Lemma 3.1: Tail bound for $k$-wise independent random variables
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• proof
• Claim 4.2
• proof