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Break-Resilient Codes with Loss Tolerance

Canran Wang, Minghui Liwang, Netanel Raviv

TL;DR

This work defines the $(t,s)$-break-resilient code (BRC) to address adversarial fragmentation and fragment loss in a unified, information-theoretic framework. It develops a Reed-Solomon–style, locator-polynomial decoding scheme that uses markers, set reconciliation of fragment substrings, and syndrome-based equations to recover missing information from unordered fragments. The authors establish fundamental redundancy lower bounds via confusability and sphere-packing arguments, and provide explicit encoding/decoding constructions achieving redundancy $\Theta(t \log^2 n + s \log n)$. The approach is applicable to diverse settings with physical fragmentation and loss (e.g., manufacturing embeds, DNA storage, and intermittent wireless links) and yields robust, provably efficient loss-tolerant break-resilient codes. The results advance the design of codes that tolerate both fragmentation and fragment loss under adversarial constraints, with practical implications for reliable data embedding and storage in nontraditional channels.

Abstract

Emerging applications in manufacturing, wireless communication, and molecular data storage require robust coding schemes that remain effective under physical distortions where codewords may be arbitrarily fragmented and partially missing. To address such challenges, we propose a new family of error-correcting codes, termed $(t,s)$-break-resilient codes ($(t,s)$-BRCs). A $(t,s)$-BRC guarantees correct decoding of the original message even after up to~$t$ arbitrary breaks of the codeword and the complete loss of some fragments whose total length is at most~$s$. This model unifies and generalizes previous approaches, extending break-resilient codes (which handle arbitrary fragmentation without fragment loss) and deletion codes (which correct bit losses in unknown positions without fragmentation) into a single information-theoretic framework. We develop a theoretical foundation for $(t,s)$-BRCs, including a formal adversarial channel model, lower bounds on the necessary redundancy, and explicit code constructions that approach these bounds.

Break-Resilient Codes with Loss Tolerance

TL;DR

This work defines the -break-resilient code (BRC) to address adversarial fragmentation and fragment loss in a unified, information-theoretic framework. It develops a Reed-Solomon–style, locator-polynomial decoding scheme that uses markers, set reconciliation of fragment substrings, and syndrome-based equations to recover missing information from unordered fragments. The authors establish fundamental redundancy lower bounds via confusability and sphere-packing arguments, and provide explicit encoding/decoding constructions achieving redundancy . The approach is applicable to diverse settings with physical fragmentation and loss (e.g., manufacturing embeds, DNA storage, and intermittent wireless links) and yields robust, provably efficient loss-tolerant break-resilient codes. The results advance the design of codes that tolerate both fragmentation and fragment loss under adversarial constraints, with practical implications for reliable data embedding and storage in nontraditional channels.

Abstract

Emerging applications in manufacturing, wireless communication, and molecular data storage require robust coding schemes that remain effective under physical distortions where codewords may be arbitrarily fragmented and partially missing. To address such challenges, we propose a new family of error-correcting codes, termed -break-resilient codes (-BRCs). A -BRC guarantees correct decoding of the original message even after up to~ arbitrary breaks of the codeword and the complete loss of some fragments whose total length is at most~. This model unifies and generalizes previous approaches, extending break-resilient codes (which handle arbitrary fragmentation without fragment loss) and deletion codes (which correct bit losses in unknown positions without fragmentation) into a single information-theoretic framework. We develop a theoretical foundation for -BRCs, including a formal adversarial channel model, lower bounds on the necessary redundancy, and explicit code constructions that approach these bounds.
Paper Structure (8 sections, 8 theorems, 39 equations, 1 figure, 1 algorithm)

This paper contains 8 sections, 8 theorems, 39 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Problem Definition
  4. Bounds
  5. Break-Resilient Codes with Loss Tolerance
  6. Encoding
  7. Decoding
  8. Redundancy Analysis

Key Result

Lemma 1

Let $\mathcal{C}\subseteq{\{0,1\}}^n$ be a $(t,s)$-BRC, and let $\mathcal{C}=\mathcal{D}_1\cup\mathcal{D}_2\cup\ldots,\cup,\mathcal{D}_{ {\lceil n/(s+1)\rceil}}$ be its partition to subcodes, where Then, the minimum Hamming distance of each $\mathcal{D}_i$

Figures (1)

  • Figure 1: Illustration of the merge operations for the case $t = 6$ and sufficiently large $s$, where the colored substrings indicate the substrings to be knocked out. (a) Initially, the baseline procedure involves knocking out all different bits between $\mathbf{x}$ and $\mathbf{y}$, which requires $12 > 6$ breaks and thus exceeds the $t$-constraint. (b) The result of applying a merge operation to $\mathbf{c}_3$, the shortest among the $\mathbf{c}_i$'s, reducing the number of required breaks to $10$ at the price of increasing the number of omitted bits by $|\mathbf{c}_3|$. (c) The result of applying a merge operation to $\mathbf{c}_5$, the second shortest among the $\mathbf{c}_i$'s, reducing the number of required breaks to $8$ at the price of increasing the number of omitted bits by $|\mathbf{c}_5|$. (d) The result of applying a merge operation to $\mathbf{c}_2$, the third shortest among the $\mathbf{c}_i$'s, reducing the number of required breaks to $6$ at the price of increasing the number of omitted bits by $|\mathbf{c}_2|$.

Theorems & Definitions (18)

  • Example 1
  • Definition 1: $(t,s)$-confusable
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 2
  • Remark 1
  • ...and 8 more