Reed-Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for All Rates Below Capacity
arXiv:2110.14631 · doi:10.1109/TIT.2023.3286452
Abstract
This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.
arXiv comment30 pages, minor changes (including title) from previous version to correct typos and clarify some statements, accepted to the IEEE Transactions on Information Theory
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