Error-Correcting Codes for the Sum Channel

2026-01-15

AI Summary

This work addresses the problem of reliable coding under a novel “sum channel” model arising in distributed and DNA storage systems, focusing on scenarios involving multiple deletions and substitution errors. The paper proposes the first explicit code capable of correcting two deletions in this setting, achieving redundancy of \(2\lceil\log_2\log_2 n\rceil + O(\ell^2)\), which is near-optimal when \(\ell=2\). Additionally, it constructs an almost information-theoretically optimal single-substitution-correcting code requiring only \(\lceil\log_2(\ell+1)\rceil\) bits of redundancy—just one bit above the theoretical lower bound. These constructions leverage combinatorial coding techniques, logarithmic redundancy analysis, and parity-check row structures to enable efficient error correction while significantly reducing redundancy overhead.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Information Theory

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We introduce the sum channel, a new channel model motivated by applications in distributed storage and DNA data storage. In the error-free case, it takes as input an $\ell$-row binary matrix and outputs an $(\ell+1)$-row matrix whose first $\ell$ rows equal the input and whose last row is their parity (sum) row. We construct a two-deletion-correcting code with redundancy $2\lceil\log_2\log_2 n\rceil + O(\ell^2)$ for $\ell$-row inputs. When $\ell=2$, we establish an upper bound of $\lceil\log_2\log_2 n\rceil + O(1)$, implying that our redundancy is optimal up to a factor of 2. We also present a code correcting a single substitution with $\lceil \log_2(\ell+1)\rceil$ redundant bits and prove that it is within one bit of optimality.

Problem

• sum channel
• error-correcting codes
• deletion errors
• substitution errors
• distributed storage

Innovation

• sum channel
• deletion-correcting codes
• substitution error correction
• redundancy optimization
• DNA data storage