Optimal Achievable Rates for Computation With Random Homologous Codes
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Building on the framework of nested coset codes by Padakandla and Pradhan, and by Lim, Feng, Pastore, Nazer, and Gastpar, the optimal rate region for computing a linear combination of sources over a multiple access channel is studied. Inner and outer bounds on this optimal rate region 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 based on joint typicality encoding. When the desired linear combination is "matched" to the structure of the multiple access channel, the inner and outer bounds coincide. This result indicates that existing coding schemes for computation based on random homologous code ensembles cannot be improved by using more powerful decoders, such as the maximum likelihood decoder.
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