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On Codes for the Noisy Substring Channel

Yonatan Yehezkeally, Nikita Polyanskii

TL;DR

This work addresses coding for a noisy substring channel where observations arise from the multisets of substrings of a string. By analyzing two noise models—substitutions and deletions—it derives sublinear redundancy bounds and, in several regimes, achieves rate-1 even under imperfect substring observations. The authors provide an explicit encoder for resilient-repeat-free sequences, propose a framework to couple these codes with classical error-correcting codes, and connect the coding problem to secondary-structure avoidance in DNA to support robust in-vivo DNA storage. The results combine probabilistic (Lovász Local Lemma) and constructive techniques to yield practical encoders and insights into when asymptotic rate costs are avoidable in these constrained observation models.

Abstract

We consider the problem of coding for the substring channel, in which information strings are observed only through their (multisets of) substrings. Due to existing DNA sequencing techniques and applications in DNA-based storage systems, interest in this channel has renewed in recent years. In contrast to existing literature, we consider a noisy channel model where information is subject to noise before its substrings are sampled, motivated by in-vivo storage. We study two separate noise models, substitutions or deletions. In both cases, we examine families of codes which may be utilized for error-correction and present combinatorial bounds on their sizes. Through a generalization of the concept of repeat-free strings, we show that the added required redundancy due to this imperfect observation assumption is sublinear, either when the fraction of errors in the observed substring length is sufficiently small, or when that length is sufficiently long. This suggests that no asymptotic cost in rate is incurred by this channel model in these cases. Moreover, we develop an efficient encoder for such constrained strings in some cases. Finally, we show how a similar encoder can be used to avoid formation of secondary-structures in coded DNA strands, even when accounting for imperfect structures.

On Codes for the Noisy Substring Channel

TL;DR

This work addresses coding for a noisy substring channel where observations arise from the multisets of substrings of a string. By analyzing two noise models—substitutions and deletions—it derives sublinear redundancy bounds and, in several regimes, achieves rate-1 even under imperfect substring observations. The authors provide an explicit encoder for resilient-repeat-free sequences, propose a framework to couple these codes with classical error-correcting codes, and connect the coding problem to secondary-structure avoidance in DNA to support robust in-vivo DNA storage. The results combine probabilistic (Lovász Local Lemma) and constructive techniques to yield practical encoders and insights into when asymptotic rate costs are avoidable in these constrained observation models.

Abstract

We consider the problem of coding for the substring channel, in which information strings are observed only through their (multisets of) substrings. Due to existing DNA sequencing techniques and applications in DNA-based storage systems, interest in this channel has renewed in recent years. In contrast to existing literature, we consider a noisy channel model where information is subject to noise before its substrings are sampled, motivated by in-vivo storage. We study two separate noise models, substitutions or deletions. In both cases, we examine families of codes which may be utilized for error-correction and present combinatorial bounds on their sizes. Through a generalization of the concept of repeat-free strings, we show that the added required redundancy due to this imperfect observation assumption is sublinear, either when the fraction of errors in the observed substring length is sufficiently small, or when that length is sufficiently long. This suggests that no asymptotic cost in rate is incurred by this channel model in these cases. Moreover, we develop an efficient encoder for such constrained strings in some cases. Finally, we show how a similar encoder can be used to avoid formation of secondary-structures in coded DNA strands, even when accounting for imperfect structures.

Paper Structure

This paper contains 9 sections, 19 theorems, 77 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works and main contribution
  3. Preliminaries
  4. Substitution noise
  5. Rate of resilient-repeat-free strings
  6. Encoding resilient-repeat-free codes
  7. Error-correcting codes for the noisy substring channel
  8. Deletion noise
  9. Secondary structure avoidance

Key Result

Lemma 1

Take $\ell\leqslant n$ and an observable pair $(I,J)\in \binom{[n]}{\ell}^2$. Further, let ${\boldsymbol x}\in\Sigma^n$ be chosen uniformly at random. Then ${\boldsymbol u}_{I,J}$ is distributed uniformly and mutually independent of $\mathset*{{\boldsymbol u}_{P,Q}}{(P,Q)\not\in \Gamma}_I$.

Figures (2)

  • Figure 1: Formation of a hairpin-loop secondary structure in an oligonucleotide.
  • Figure 2: Imperfect stem in a hairpin-loop structure.

Theorems & Definitions (30)

  • Example 1
  • Definition 1
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Definition 2
  • Example 2
  • Theorem 2
  • Lemma 2
  • Corollary 2
  • ...and 20 more