We generalize the problem of recovering a lost/erased symbol in a Reed-Solomon code to the scenario in which some side information about the lost symbol is known. The side information is represented as a set $S$ of linearly independent combinations of the sub-symbols of the lost symbol. When $S = \varnothing$, this reduces to the standard problem of repairing a single codeword symbol. When $S$ is a set of sub-symbols of the erased one, this becomes the repair problem with partially lost/erased symbol. We first establish that the minimum repair bandwidth depends on $|S|$ and not the content of $S$ and construct a lower bound on the repair bandwidth of a linear repair scheme with side information $S$. We then consider the well-known subspace-polynomial repair schemes and show that their repair bandwidths can be optimized by choosing the right subspaces. Finally, we demonstrate several parameter regimes where the optimal bandwidths can be achieved for full-length Reed-Solomon codes.

翻译：我们推广了里德-所罗门码中丢失/擦除符号的恢复问题，将其扩展到已知丢失符号部分边信息的场景。该边信息表示为丢失符号子符号的一组线性无关组合构成的集合 $S$。当 $S = \varnothing$ 时，该问题退化为标准单码字符号修复问题；当 $S$ 为擦除符号的子符号集时，则成为部分丢失/擦除符号的修复问题。我们首先证明最小修复带宽仅取决于 $|S|$ 而非 $S$ 的具体内容，并构建了带边信息 $S$ 的线性修复方案修复带宽下界。随后考虑已知的子空间多项式修复方案，表明可通过选择恰当的子空间优化其修复带宽。最终，我们展示了在若干参数范围内，全长度里德-所罗门码可实现最优带宽。