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Capacity-Achieving Codes for Noisy Insertion Channels

Hengfeng Liu, Chunming Tang, Cuiling Fan

Abstract

DNA storage has emerged as a promising solution for large-scale and long-term data preservation. Among various error types, insertions are the most frequent errors occurring in DNA sequences, where the inserted symbol is often identical or complementary to the original, and in practical implementations, noise can further cause the inserted symbol to mutate into a random one, which creates significant challenges to reliable data recovery. In this paper, we investigate a new noisy insertion channel, where infinitely many insertions of symbols complement or identical to the original ones and up to one insertion of random symbol may occur. We determine the coding capacity of the noisy channel and construct asymptotically optimal error-correcting codes achieving the coding capacity.

Capacity-Achieving Codes for Noisy Insertion Channels

Abstract

DNA storage has emerged as a promising solution for large-scale and long-term data preservation. Among various error types, insertions are the most frequent errors occurring in DNA sequences, where the inserted symbol is often identical or complementary to the original, and in practical implementations, noise can further cause the inserted symbol to mutate into a random one, which creates significant challenges to reliable data recovery. In this paper, we investigate a new noisy insertion channel, where infinitely many insertions of symbols complement or identical to the original ones and up to one insertion of random symbol may occur. We determine the coding capacity of the noisy channel and construct asymptotically optimal error-correcting codes achieving the coding capacity.

Paper Structure

This paper contains 5 sections, 11 theorems, 52 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Codes for the Noisy Insertion Channel
  4. Decoding Algorithm and Complexity
  5. Conclusion

Key Result

Theorem 1

Let $x=x_0x_1\dots x_{n-1}\in \mathbb{Z}_q^{*}$ be an arbitrary sequence. Then $x$ is irreducible if and only if $x_i\notin \{x_{i+1}, \overline{x_{i+1}} \}$ for all $i=0,1, \dots, n-1$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (33)

  • Definition 1
  • Example 1
  • Example 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 3
  • Theorem 1
  • proof
  • ...and 23 more