{
  "abstract": "This mostly expository note studies the sample-noise regimes in which the testing problem for noisy $k$-XOR in dimension $n$ can be efficiently solved. Concretely, we ask what pairs $(β, δ)$ admit algorithms running in time $n^{O(k)}$ where $m = n^β, β\\in [0,k]$ is the number of samples and $\\mathsf{Bern}(n^{-δ}),δ\\in (0,+\\infty)$ is the noise distribution. The current state-of-the-art algorithms are: I. A folklore algorithm when $β> 1, δ>1$ based on Gaussian elimination; II. A folklore algorithm when $β\\ge k/2$ based on the birthday paradox; III. A recent algorithm in the regime $β> δ+ k(1-δ), δ\\in (0,1)$ based on a reduction to the first algorithm [DJ24,CSZ24]. IV. Extending [AF09], we strictly improve the last regime to $β> δ+ (k-1)(1-δ), δ\\in (0,1).$ These algorithms correspond to efficiently finding short cycles in uniform hypergraphs and can be equivalently phrased as algorithmic Moore bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27147",
  "author_total": 1,
  "authors": [
    {
      "name": "Kiril Bangachev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/32s1dfrtxzqdm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27147",
  "fulltext_url": null,
  "id": "32s1dfrtxzqdm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27147",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7223108439",
  "pdf_url": "https://arxiv.org/pdf/2610.27147",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/32s1dfrtxzqdm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27147",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27147"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "A Short Note on Algorithms For The Noisy $k$-XOR Problem",
  "title_text": "A Short Note on Algorithms For The Noisy k-XOR Problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/32s1dfrtxzqdm",
  "venue": "arXiv",
  "versions": []
}