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Error correction methods based on two-faced processes

Boris Ryabko

TL;DR

The paper introduces a novel error-correction approach that transforms input data into two-faced processes to create strong intersymbol dependence, enabling error reduction on erasure and binary-symmetric channels. It provides a constructive coding framework based on two-faced processes and transformations, with Code_EC and Code_BSC achieving linear-time encoding/decoding and giving BER bounds that scale as BER ~ pi o(pi) for small channel error probability. The analysis shows that increasing memory in the two-faced process reduces the uncorrected error frequency, and that an appropriate choice of parameters l and w yields provable improvements. The work situates these codes within Shannon theory, noting that zero error is possible when source entropy is below channel capacity and suggesting compression or ML decoding as future directions to approach fundamental limits.

Abstract

A new approach to the problem of error correction in communication channels is proposed, in which the input sequence is transformed in such a way that the interdependence of symbols is significantly increased. Then, after the sequence is transmitted over the channel, this property is used for error correction so that the remaining error rate is significantly reduced. The complexity of encoding and decoding is linear.

Error correction methods based on two-faced processes

TL;DR

The paper introduces a novel error-correction approach that transforms input data into two-faced processes to create strong intersymbol dependence, enabling error reduction on erasure and binary-symmetric channels. It provides a constructive coding framework based on two-faced processes and transformations, with Code_EC and Code_BSC achieving linear-time encoding/decoding and giving BER bounds that scale as BER ~ pi o(pi) for small channel error probability. The analysis shows that increasing memory in the two-faced process reduces the uncorrected error frequency, and that an appropriate choice of parameters l and w yields provable improvements. The work situates these codes within Shannon theory, noting that zero error is possible when source entropy is below channel capacity and suggesting compression or ML decoding as future directions to approach fundamental limits.

Abstract

A new approach to the problem of error correction in communication channels is proposed, in which the input sequence is transformed in such a way that the interdependence of symbols is significantly increased. Then, after the sequence is transmitted over the channel, this property is used for error correction so that the remaining error rate is significantly reduced. The complexity of encoding and decoding is linear.
Paper Structure (10 sections, 3 theorems, 24 equations)

This paper contains 10 sections, 3 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The main tool of the proposed method: two-faced processes
  3. Intuitive description of the ideas underlying the proposed method
  4. Two-faced processes and transformations
  5. two-faced processes
  6. two-faced transformations
  7. The code
  8. Erasure channel
  9. BSC channel
  10. Conclusion

Key Result

Theorem 1

Let $l > 0$ be an integer, $w$ be a uniformly distributed binary word of $l$ letters, and let $x=x_1...x_n$ be generated randomly according to a Bernoulli process with parameters $p \in (0,1)$. Then the sequence $\varphi_{l,u}(x)$ obeys distribution of two-faced process $T_l$ of order $l$ (with matr

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • proof
  • Claim 1
  • Theorem 3
  • proof