Error-Correcting Codes for the Sum Channel
For researchers in coding theory and data storage, this provides new code constructions and bounds for a novel channel model, though the results are incremental as they extend known techniques.
The paper introduces the sum channel model for distributed and DNA storage, and constructs a two-deletion-correcting code with redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) for ℓ-row inputs, which is optimal up to a factor of 2 for ℓ=2. It also presents a single-substitution-correcting code with ⌈log₂(ℓ+1)⌉ redundant bits, within one bit of optimality.
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.