Error-Correcting Codes for the Sum Channel

TL;DR

This paper introduces the sum channel model and develops error-correcting codes for it, achieving near-optimal redundancy for deletion and substitution errors in distributed storage applications.

Contribution

The paper constructs new error-correcting codes for the sum channel, including two-deletion and single substitution correction codes, with redundancy close to theoretical limits.

Findings

Constructed a two-deletion-correcting code with redundancy $2\lceil ext{log}_2 ext{log}_2 n ceil + O( ext{ell}^2)$.

Established an upper bound of $ ext{ceil}( ext{log}_2 ext{log}_2 n)+ O(1)$ for $ ext{ell}=2$, showing near-optimality.

Presented a code correcting a single substitution with redundancy within one bit of optimality.

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.