Near Optimal Code Construction for the Adversarial Torn Paper Channel With Edit Errors

TL;DR

This work studies the adversarial torn paper channel with up to $t_e$ edit errors that precedes the division of the word into $t+1$ fragments. It develops near-optimal coding schemes by extending the Raviv et al. backbone with markers, mutually uncorrelated components, and hierarchical hashing to enable exact reconstruction from the fragments even in the presence of edits. The main results include a randomized $t$-breaks $t_e$-edit-errors resilient code ${\cal C}^{b,e}_{t,t_e}$ of length $n$ with redundancy $\Theta\left((t+t_e)\log n \log \log n\right)$ under the regime $t+t_e = o\left(\frac{n}{\log n \log \log n}\right)$, and an explicit encoder achieving redundancy $\Theta\left((t+t_e) \log m \log \frac{m}{t+t_e}\right)$ with length $n=m+O\left((t+t_e)\log m \log \log m\right)$. The paper also develops list-decoding bounds for the torn paper channel, linking adversarial and probabilistic models by quantifying the size of the potential codeword list when more than $t$ breaks occur. These results advance robust data recovery for DNA storage and related fragmentation-prone applications by formalizing near-optimal redundancy and decoding strategies under adversarial fragmentation and edit errors.

Abstract

Motivated by DNA storage systems and 3D fingerprinting, this work studies the adversarial torn paper channel with edit errors. This channel first applies at most $t_e$ edit errors (i.e., insertions, deletions, and substitutions) to the transmitted word and then breaks it into $t+1$ fragments at arbitrary positions. In this paper, we construct a near optimal error correcting code for this channel, which will be referred to as a $t$-breaks $t_e$-edit-errors resilient code. This code enables reconstructing the transmitted codeword from the $t+1$ noisy fragments. Moreover, we study list decoding of the torn paper channel by deriving bounds on the size of the list (of codewords) obtained from cutting a codeword of a $t$-breaks resilient code $t'$ times, where $t' > t$.

Figures (4)

• Figure 1: A codeword ${\boldsymbol c} \in {\cal C}^b_{t}$ according to Construction \ref{['Cons:RavivCode']} and Definitions \ref{['Def:LegitWord']}, \ref{['Def:GenerateRed']}, and \ref{['Def:RedB']}. ${\boldsymbol z}$ contains many markers. The leftmost part depicts the division to blocks in each level. ${\boldsymbol r}$ is obtained by inserting markers $M_{red}$ to ${\boldsymbol r}^A| {\boldsymbol r}^B$. Lastly, ${\boldsymbol c} = {\boldsymbol z}| {\boldsymbol r}$.
• Figure 2: Redundancy ${\boldsymbol r}$ in ${\cal C}^{b,e}_{t,t_e}$.
• Figure 3: Matching between ${\boldsymbol c}$ and ${\boldsymbol c}^e$, and estimations ${\boldsymbol z}^{\ell,e}$ and ${\boldsymbol z}^{\ell+1,e}$, specifically in this case $N^{\ell}=9, \Tilde{k}=6, {\Tilde{j}}_1 =1, {\Tilde{j}}_2 = 2, {\Tilde{j}}_3 = 4, {\Tilde{j}}_4 = 7, {\Tilde{j}}_5 = 8$, and ${\Tilde{j}}_6 = 9$.
• Figure 4: $N\left({\boldsymbol f}_i,j_1 \right)=3, N \left( {\boldsymbol f}_i, j_2\right) = 2$. Moreover, $j_1 \in J \left({\boldsymbol f}_i\right)$ as half of the $5$ traversed hash values match, while $j_2 \notin J \left({\boldsymbol f}_i\right)$.

Theorems & Definitions (4)

• proof : Sketch proof of Lemma \ref{['lem:Decoding']}
• Claim 1
• Claim 2
• proof : Sketch proof of Theorem \ref{['theorem:ExplicitCode']}