The problem of computing a linear combination of sources over a multiple access channel is studied. Inner and outer bounds on the optimal tradeoff between the communication rates are established when encoding is restricted to random ensembles of homologous codes, namely, structured nested coset codes from the same generator matrix and individual shaping functions, but when decoding is optimized with respect to the realization of the encoders. For the special case in which the desired linear combination is “matched” to the structure of the multiple access channel in a natural sense, these inner and outer bounds coincide. This result indicates that most, if not all, coding schemes for computation in the literature that rely on random construction of nested coset codes cannot be improved by using more powerful decoders such as the maximum likelihood decoder. The proof techniques are adapted to characterize the rate region for broadcast channels achieved by Marton’s (random) coding scheme under maximum likelihood decoding. By generalizing some of the techniques, a single-letter outer bound for the capacity region of the computation problem is presented and compared with the inner bound achieved by homologous codes.

中文翻译：

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关于随机同源码线性计算的最优可实现率

研究了在多路访问信道上计算源的线性组合的问题。当编码仅限于同源代码的随机集合时，通信速率之间的最佳权衡的内界和外界就建立起来了，即来自相同生成器矩阵和单个整形函数的结构化嵌套陪集代码，但是当解码相对于编码器的实现。对于期望的线性组合在自然意义上与多址信道的结构“匹配”的特殊情况，这些内边界和外边界重合。该结果表明，大多数（如果不是全部）文献中依赖于随机构造嵌套陪集代码的计算编码方案无法通过使用更强大的解码器（例如最大似然解码器）来改进。证明技术适用于在最大似然解码下表征由 Marton 的（随机）编码方案实现的广播信道的速率区域。通过推广一些技术，提出了计算问题容量区域的单字母外界，并与同源代码实现的内界进行了比较。