{
  "abstract": "The notion of sum-freedom of binary functions was introduced recently by C. Carlet as a generalization of the APN functions used in cryptography; the $q$-ary version of the notion is a natural extension. For each integer $k$ with $0\\le k\\le n$, there is a $k$th order sum-free function on $\\Bbb F_{2^n}$. It is also known that when $k/n$ is not close to 0 or 1, the multiplicative inverse function on $\\Bbb F_{2^n}$ is not $k$th order sum-free. We generalize these two results to $q$-ary functions. APN functions have a coding theoretic characterization. We generalize the characterization to sum-free functions of arbitrary order over any finite field. It is well known that the Welch functions is 2nd order sum-free. We give an alternative proof for this result which leads to a more general algebraic question. We also investigate that the 3rd order sum-freedom of the Welch function and power functions of algebraic degree 3. We formulate a conjecture about the 3rd order sum-freedom of the Welch function which is supported by strong numerical evidence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31489",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiang-dong Hou"
    },
    {
      "name": "Shujun Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/az0z987ns6384.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31489",
  "fulltext_url": null,
  "id": "az0z987ns6384",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31489",
  "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",
      "math.NT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214583763",
  "pdf_url": "https://arxiv.org/pdf/2609.31489",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/az0z987ns6384.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2609.31489",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31489"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Further Results on Sum-Freedom of Binary and $q$-ary Functions",
  "title_text": "Further Results on Sum-Freedom of Binary and q-ary Functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/az0z987ns6384",
  "venue": "arXiv",
  "versions": []
}