{
  "abstract": "We compare the performance of the RPA and CHIRRUP decoding algorithms for second-order Reed--Muller codes. The RPA algorithm recovers a binary quadratic form associated with a skew-symmetric matrix $M$, whereas CHIRRUP recovers a $\\mathbb{Z}_4$-valued quadratic form associated with a symmetric matrix $P$. We describe how the Gray map connects the evaluation vectors of these forms, inducing a rank-preserving correspondence between $M$ and $P$. We analyze how Euclidean distance between evaluation vectors is governed by matrix rank differences using Delsarte--Goethals sets $DG(m,r)$. We demonstrate that CHIRRUP uses repeated measurements more effectively than RPA on these ensembles. While a kernel-aware variant of RPA improves performance on singular quadratic forms by aggregating projection directions across cosets of $\\ker M$, CHIRRUP's tree search combines multiple row estimates more efficiently on $DG(m,r)$ when $r < \\frac{m-1}{2}$, outperforming vanilla RPA.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24647",
  "author_total": 2,
  "authors": [
    {
      "name": "Ziang Gao"
    },
    {
      "name": "Robert Calderbank"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f6nbzhknp3z00.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24647",
  "fulltext_url": null,
  "id": "f6nbzhknp3z00",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24647",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214009150",
  "pdf_url": "https://arxiv.org/pdf/2609.24647",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f6nbzhknp3z00.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24647",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24647"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes",
  "title_text": "The Value of Duplicate Measurements When Decoding Second-Order Reed-Muller Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/f6nbzhknp3z00",
  "venue": "arXiv",
  "versions": []
}