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Error-Resilient Weakly Constrained Coding via Row-by-Row Coding

Prachi Mishra, Navin Kashyap

TL;DR

This work addresses the vulnerability of row-by-row weakly constrained codes to errors by fixing the column-concatenation order, thereby removing the decoder’s need to infer the stitching order from the codeword. For primitive subgraphs $G$ of the first-order de Bruijn graph $D_{1,2}$ and an $n$-integral stationary Markov chain $P$, the authors show there exists a constant $Z$ (dependent only on $P$) such that one can injectively encode $m$ messages into a codeword of length $N=(m+Z)n+1$ with exact edge counts $P(e)(N-1)$, where the top row is $U_ ho$ and the bottom row is the left-shifted $U_ ho$, enabling fixed stitching. The main technical contributions are a two-step transition method and a $1$-$1$ boosting technique, which together guarantee the transition from the top to the bottom row in a bounded number of steps, independent of $n$, while maintaining the weak constraint. This yields improved error resilience and reduced redundancy compared to prior weakly constrained schemes, with future work aimed at extending to higher-order de Bruijn graphs. The approach holds promise for robust DNA-based storage and related applications where pattern-frequency constraints must be maintained under errors.

Abstract

A weakly constrained code is a collection of finite-length strings over a finite alphabet in which certain substrings or patterns occur according to some prescribed frequencies. Buzaglo and Siegel (ITW 2017) gave a construction of weakly constrained codes based on row-by-row coding, that achieved the capacity of the weak constraint. In this paper, we propose a method to make this row-by-row coding scheme resilient to errors.

Error-Resilient Weakly Constrained Coding via Row-by-Row Coding

TL;DR

This work addresses the vulnerability of row-by-row weakly constrained codes to errors by fixing the column-concatenation order, thereby removing the decoder’s need to infer the stitching order from the codeword. For primitive subgraphs of the first-order de Bruijn graph and an -integral stationary Markov chain , the authors show there exists a constant (dependent only on ) such that one can injectively encode messages into a codeword of length with exact edge counts , where the top row is and the bottom row is the left-shifted , enabling fixed stitching. The main technical contributions are a two-step transition method and a - boosting technique, which together guarantee the transition from the top to the bottom row in a bounded number of steps, independent of , while maintaining the weak constraint. This yields improved error resilience and reduced redundancy compared to prior weakly constrained schemes, with future work aimed at extending to higher-order de Bruijn graphs. The approach holds promise for robust DNA-based storage and related applications where pattern-frequency constraints must be maintained under errors.

Abstract

A weakly constrained code is a collection of finite-length strings over a finite alphabet in which certain substrings or patterns occur according to some prescribed frequencies. Buzaglo and Siegel (ITW 2017) gave a construction of weakly constrained codes based on row-by-row coding, that achieved the capacity of the weak constraint. In this paper, we propose a method to make this row-by-row coding scheme resilient to errors.
Paper Structure (19 sections, 3 theorems, 16 equations, 4 figures, 5 tables, 2 algorithms)

This paper contains 19 sections, 3 theorems, 16 equations, 4 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov Chains
  4. de Bruijn Graphs
  5. Row-By-Row Constrained Coding
  6. Weakly Constrained Coding
  7. Error-Resilient Weakly Constrained Coding
  8. Our Main Results
  9. Proof Sketch for Theorem \ref{['thm:Z']}
  10. Two-step transition
  11. $1$--$1$ boosting
  12. Concluding Remarks
  13. Row-By-Row Constrained Coding
  14. Incorrect Payload extraction
  15. Proof details
  16. ...and 4 more sections

Key Result

Proposition 1

If for all $e\in E, P(e)n \geq |V|$ holds, then there exists a constant $b$ that depends only on $G$ and the Markov chain $P$, such that one can encode $m$ messages from $C_{\mathrm{rbr}}$ into a length $N=(b\log n +m+N_{G})n +k$ codeword $w\in S(G)$, in which every pattern $e\in E$ occurs exactly $

Figures (4)

  • Figure 1: The de Bruijn graph $D_{1,2}$.
  • Figure 2: A $(G,P,n)$-array of size $(1+m+Z) \times n$, with first row $U_\pi$, $m$ rows of payload, and last row $\sigma_{\mathrm{left}}(U_\pi)$.
  • Figure 3: The graph $G=(V,E,L)$
  • Figure :

Theorems & Definitions (9)

  • Proposition 1: Corollary 1 in buzaglo2017weakly
  • Example 1
  • Theorem 1
  • Corollary 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6