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Error Correction for Message Streams

Meghal Gupta, Rachel Yun Zhang

TL;DR

The paper addresses computing functions from a binary message stream under adversarial noise while using sublinear space. It introduces a noise-resilient encoding enc that expands the stream to m bits and a decoder that converts any deterministic low-space streaming algorithm into a robust, low-space counterpart on the encoded stream, tolerating up to a (1/4 − ε) fraction of errors. The construction leverages locally decodable codes with confidence measures and a recursive decomposition that aggregates decoding attempts while keeping memory usage polylogarithmic in n; for linear functions the encoding length reduces to near-quadratic, and for general functions to near-quadratic or near-linear in refined variants. The approach yields practical guarantees: high-probability correctness, explicit encoding, and sublinear space overhead, offering a framework bridging streaming algorithms, error-correcting codes, and interactive coding concepts. The results have significance for robust streaming in resource-constrained devices and noisy communication settings, with potential improvements and refinements explored in subsequent work.

Abstract

In the setting of error correcting codes, Alice wants to send a message $x \in \{0,1\}^n$ to Bob via an encoding $\text{enc}(x)$ that is resilient to error. In this work, we investigate the scenario where Bob is a low space decoder. More precisely, he receives Alice's encoding $\text{enc}(x)$ bit-by-bit and desires to compute some function $f(x)$ in low space. A generic error-correcting code does not accomplish this because decoding is a very global process and requires at least linear space. Locally decodable codes partially solve this problem as they allow Bob to learn a given bit of $x$ in low space, but not compute a generic function $f$. Our main result is an encoding and decoding procedure where Bob is still able to compute any such function $f$ in low space when a constant fraction of the stream is corrupted. More precisely, we describe an encoding function $\text{enc}(x)$ of length $\text{poly}(n)$ so that for any decoder (streaming algorithm) $A$ that on input $x$ computes $f(x)$ in space $s$, there is an explicit decoder $B$ that computes $f(x)$ in space $s \cdot \text{polylog}(n)$ as long as there were not more than $\frac14 - \varepsilon$ fraction of (adversarial) errors in the input stream $\text{enc}(x)$.

Error Correction for Message Streams

TL;DR

The paper addresses computing functions from a binary message stream under adversarial noise while using sublinear space. It introduces a noise-resilient encoding enc that expands the stream to m bits and a decoder that converts any deterministic low-space streaming algorithm into a robust, low-space counterpart on the encoded stream, tolerating up to a (1/4 − ε) fraction of errors. The construction leverages locally decodable codes with confidence measures and a recursive decomposition that aggregates decoding attempts while keeping memory usage polylogarithmic in n; for linear functions the encoding length reduces to near-quadratic, and for general functions to near-quadratic or near-linear in refined variants. The approach yields practical guarantees: high-probability correctness, explicit encoding, and sublinear space overhead, offering a framework bridging streaming algorithms, error-correcting codes, and interactive coding concepts. The results have significance for robust streaming in resource-constrained devices and noisy communication settings, with potential improvements and refinements explored in subsequent work.

Abstract

In the setting of error correcting codes, Alice wants to send a message to Bob via an encoding that is resilient to error. In this work, we investigate the scenario where Bob is a low space decoder. More precisely, he receives Alice's encoding bit-by-bit and desires to compute some function in low space. A generic error-correcting code does not accomplish this because decoding is a very global process and requires at least linear space. Locally decodable codes partially solve this problem as they allow Bob to learn a given bit of in low space, but not compute a generic function . Our main result is an encoding and decoding procedure where Bob is still able to compute any such function in low space when a constant fraction of the stream is corrupted. More precisely, we describe an encoding function of length so that for any decoder (streaming algorithm) that on input computes in space , there is an explicit decoder that computes in space as long as there were not more than fraction of (adversarial) errors in the input stream .
Paper Structure (34 sections, 22 theorems, 36 equations, 1 figure, 2 algorithms)

This paper contains 34 sections, 22 theorems, 36 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Model
  3. Our Results
  4. Linear Streaming Algorithms.
  5. Randomized Algorithms.
  6. Non-Binary Input Alphabets.
  7. Discussion of Subsequent Work
  8. Related Works
  9. Streaming Algorithms.
  10. Error Correcting Codes.
  11. Coding for Interactive Communication.
  12. Overview of Techniques
  13. A recursive computation.
  14. Decoding with confidences.
  15. Computing non-linear functions.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Fix $\varepsilon,\delta>0$. For any $X\subseteq \{0,1\}^n$, there is a noise resilient transformation consisting of an encoding $\mathsf{enc} : X\subseteq\{0,1\}^n \to \{0,1\}^m$, and a transformation of algorithms $B$ that is $\left( \frac{1}{4} - \varepsilon \right) \cdot m$ error resilient with p

Figures (1)

  • Figure 1: In our transformation, each sub-function is computed $\ell$ times instead of $1$, and in each of the $\ell$ chunks, the sub-functions $f_i$ are computed in according to a random permutation $\pi$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Theorem 3.2: Berlekamp-Welch; see e.g. GemmellS92
  • Theorem 3.3: GMD Decoding Forney66Guruswami06
  • Theorem 4.1
  • proof
  • Definition 5.0
  • Theorem 5.1
  • ...and 29 more