{
  "abstract": "For $n\\ge5$, let $F\\colon\\mathbb{F}_2^n\\to \\mathbb{F}_2^n$ be almost perfect nonlinear and write $N=2^n$. It is proven that the linearity $\\mathcal{L}(F)$ of $F$, i.e., the largest absolute Walsh coefficient of a nonzero component, is at most $N-6$. If $n$ is even, it is further proven that $\\mathcal{L}(F)\\le N-\\sqrt N/4-2$. These bounds improve the general upper bound of $N-4$ in odd dimension and $N-6$ in even dimensions $n \\geq 10$. It is further proven that, for each fixed $k$, the $k$-th largest absolute Walsh coefficient among nonzero components, counted with multiplicity, is at most $(1+O_k(2^{-n}))N/\\sqrt{k}$ as $n\\to\\infty$. The second and fourth largest coefficients are at most $2\\lfloor N/3\\rfloor$ and $N/2$, respectively. Finally, a bound on the linearity in terms of the number $q$ of nonplateaued nonzero components is derived. In odd dimension, for $q>0$, we have $\\mathcal{L}(F)^2\\le N(1+\\sqrt{q(N-1)})$, so that $\\mathcal{L}(F)/N\\to1$ implies $q/N\\to1$. In even dimension, for every $1/2\\le C<1$, the condition $q\\le(4C-C^2-1)N/4+1$ implies $\\mathcal{L}(F)\\le CN$. In particular, $q\\le3N/16+2$ implies $\\mathcal{L}(F)\\le N/2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35689",
  "author_total": 2,
  "authors": [
    {
      "name": "Christof Beierle"
    },
    {
      "name": "Lukas Kölsch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5svq1gspre9d7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35689",
  "fulltext_url": null,
  "id": "5svq1gspre9d7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35689",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214700164",
  "pdf_url": "https://arxiv.org/pdf/2609.35689",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5svq1gspre9d7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.35689",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35689"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Linearity bounds for APN functions",
  "title_text": "Linearity bounds for APN functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/5svq1gspre9d7",
  "venue": "arXiv",
  "versions": []
}