The Covering Radius of the Reed--Muller Code \texorpdfstring{RM(3,8)}{RM(3,8)} is 56

Abstract

We prove that the covering radius of the third-order Reed--Muller code in eight variables is fifty-six. The proof separates even- and odd-weight inputs by moment-controlled slicing. On the even side, a transport invariant on quadratic-label stars and consistency of singleton restrictions give a compatible cubic correction. On the odd side, symmetry of the third moment yields a covariance-rank obstruction. Its rank-two exception is a quartic plus three points of a two-plane; the zero-second-moment case is a prescribed-point perturbation of a quintic. We prove that every such point perturbation has third-order nonlinearity at most fifty-five by a marked derivative identity and complete restriction-code covers. Consequently the maximum among exact-degree-eight functions is fifty-five. The structural reductions are proved analytically, and complete finite certificates verify the explicitly stated local covering and compatibility assertions. The matching lower bound follows from the published quartic relative radius.

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BibTeX RIS

BibTeX

@misc{zhao2026covering,
  title = {{The Covering Radius of the Reed--Muller Code \texorpdfstring$\RM(3,8)$RM(3,8) is 56}},
  author = {Zhihang Zhao and Yansheng Wu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03186},
  eprint = {2611.03186},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03186},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - The Covering Radius of the Reed--Muller Code \texorpdfstring{RM(3,8)}{RM(3,8)} is 56
AU  - Zhihang Zhao
AU  - Yansheng Wu
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03186
UR  - https://arxiv.org/abs/2611.03186
AB  - We prove that the covering radius of the third-order Reed--Muller code in eight variables is fifty-six. The proof separates even- and odd-weight inputs by moment-controlled slicing. On the even side, a transport invariant on quadratic-label stars and consistency of singleton restrictions give a compatible cubic correction. On the odd side, symmetry of the third moment yields a covariance-rank obstruction. Its rank-two exception is a quartic plus three points of a two-plane; the zero-second-moment case is a prescribed-point perturbation of a quintic. We prove that every such point perturbation has third-order nonlinearity at most fifty-five by a marked derivative identity and complete restriction-code covers. Consequently the maximum among exact-degree-eight functions is fifty-five. The structural reductions are proved analytically, and complete finite certificates verify the explicitly stated local covering and compatibility assertions. The matching lower bound follows from the published quartic relative radius.
ER  -

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