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.

翻译：本文提出求和信道这一新型信道模型，其设计灵感来源于分布式存储与DNA数据存储的应用需求。在无差错情况下，该信道以$\ell$行二进制矩阵作为输入，输出$(\ell+1)$行矩阵，其中前$\ell$行与输入相同，末行为其奇偶校验（求和）行。针对$\ell$行输入，我们构建了冗余度为$2\lceil\log_2\log_2 n\rceil + O(\ell^2)$的双删除纠错码。当$\ell=2$时，我们证明了$\lceil\log_2\log_2 n\rceil + O(1)$的冗余度上界，表明所构建码的冗余度在2倍因子内达到最优。此外，我们提出了一种可纠正单比特替换的编码方案，其冗余位数为$\lceil \log_2(\ell+1)\rceil$，并证明该方案与最优解至多相差1比特。