Error-Correcting Codes for the Sum Channel
Lyan Abboud, Eitan Yaakobi
TL;DR
This work introduces the sum channel, where an input matrix $\boldsymbol{X}$ is extended by a parity row to form $\boldsymbol{X}^+$. It develops near-optimal codes that correct two deletions with redundancy $2\lceil\log_2\log_2 n\rceil + O(\ell^2)$ for any $\ell$, and proves an order-optimal lower bound for $\ell=2$ via a clique-cover argument. It then extends to arbitrary $\ell$ using a tensor-product construction based on SVT variants and per-row syndromes, achieving the same redundancy order. Separately, it gives a simple optimal $(\ell;1)_{\SID}$-correcting code with redundancy $\lceil\log_2(\ell+1)\rceil$, and shows a near-tight upper bound on the size of $(2;2)_{\mathbb{D}}$ codes via a detailed clique-cover analysis. Overall, the paper provides concrete, low-redundancy coding schemes for the sum channel applicable to RAID-like storage and DNA data storage scenarios, along with fundamental limits for these codes.
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.