Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes
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| Format: | Preprint |
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2025
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| _version_ | 1866915736529666048 |
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| author | Ly, Hoang Soljanin, Emina |
| author_facet | Ly, Hoang Soljanin, Emina |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes Ly, Hoang Soljanin, Emina Information Theory Combinatorics 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. |
| title | Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes |
| topic | Information Theory Combinatorics |
| url | https://arxiv.org/abs/2508.08736 |