The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability.

翻译：Reed 为阶数为 r、长度为 2^m 的 Reed-Muller 码 RM(r, m) 提出的经典大数逻辑译码器，需要 r+1 个顺序步骤，从最高次到最低次依次译码信息符号。后续的几种译码算法减少了步骤数，但仅限于特定参数集，或是以性能下降为代价，或依赖于某些组合结构的存在性。我们证明，对于所有 r 和 m 的取值，任何一步大数逻辑译码器——即所有大数表决在一个步骤中同时执行、无需顺序处理的译码器——最多只能纠正 d_min/4 个错误，其中 d_min 表示码的最小距离。随后，我们提出了一种新的硬判决译码器，它在单步内完成译码并达到了这一纠错极限。该译码器适用于所有 r 和 m，可视为 Reed 原始算法的并行实现，能够同时译码所有信息符号。值得注意的是，我们还证明了该译码器在删除信道下是最优的：它能够从最多 d_min-1 个符号的任意删除模式中恢复信息——这是理论极限。据我们所知，这是首个同时实现最优删除纠正和最大一步纠错能力的 RM 码一步译码器。