{
  "abstract": "The cross-correlation problem is a classical problem in sequence design. In this paper, we determine the cross-correlation distribution of the Niho-type decimation $d=4(2^m-1)+1$ over $\\mathbb F_{2^{2m}}$ for every positive integer $m$. With $q=2^m$, this is equivalent to determining the distribution of the number of roots in $U_{q+1}$ of $f_a(x)=x^7+ax^4+\\bar a x^3+1$ for $a\\in\\mathbb F_{q^2}$, where $U_{q+1}=\\{x\\in\\mathbb F_{q^2}:x^{q+1}=1\\}$. The main difficulty is to count the four-element subsets of $U_{q+1}$ that may occur as root sets of $f_a$. We normalize such subsets by their product and study the resulting condition through the associated resolvent. This reduces the required enumeration to several equations over $\\mathbb F_q$, which are evaluated by character sums and Kloosterman sums. The remaining mixed Kloosterman sum is related to a Kloosterman sum over $\\mathbb F_{2^{2m}}$ and is evaluated using a theorem of Carlitz. Consequently, we obtain explicit formulas for all the frequencies in the cross-correlation distribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04683",
  "author_total": 2,
  "authors": [
    {
      "name": "Maosheng Xiong"
    },
    {
      "name": "Haode Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afn1970a0s7fn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04683",
  "fulltext_url": null,
  "id": "afn1970a0s7fn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04683",
  "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": "W7211889705",
  "pdf_url": "https://arxiv.org/pdf/2609.04683",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afn1970a0s7fn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.04683",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04683"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "The Cross-Correlation Distribution of the Niho-Type Decimation $d=4(2^m-1)+1$",
  "title_text": "The Cross-Correlation Distribution of the Niho-Type Decimation d=4(2^m−1)+1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/afn1970a0s7fn",
  "venue": "arXiv",
  "versions": []
}