{
  "abstract": "We give a uniform, explicit and elementary recursive construction of two-dimensional arrays with perfect (ideal) periodic autocorrelation over a \\emph{fixed} alphabet of $K$ roots of unity and \\emph{unbounded} size: for every integer $K\\ge2$ and every $n\\ge1$ a perfect $K^n\\times K^n$ array whose entries are $K$-th roots of unity, given by a nested sum of floor terms. The construction works with \\emph{twisted} (quasi-perfect) arrays indexed by a type $v\\in\\Z_K^2$, of which ordinary perfect arrays are the type $v=0$ members. For $K=2$ the types are Jedwab's generalised perfect binary arrays and the recursion specialises to the doubling constructions of Jedwab--Mitchell and Wild; for prime $K$ the arrays are the relative difference sets whose existence in abelian $p$-groups was settled by Ma and Schmidt; and the bare existence of perfect $K^n\\times K^n$ arrays over $K$ roots of unity follows, for every $K$, from recent constructions of group-invariant Butson Hadamard matrices. The contribution of this note is therefore not existence but the calculus: types with arbitrary entries in $\\Z_K^2$; three spectral lemmas (a twisted periodic extension of a type-$v$ array has a flat spectrum supported on one coset of $K\\Z^2$; interleaving arrays whose spectra are flat on disjoint cosets gives a perfect array; and a shear of the index plane with a floor-function carry permutes types through the action of $SL_2(\\Z_K)$ on $\\Z_K^2$, whose orbits are the classes $\\gcd(v_1,v_2,K)=g$); a one-directional lift that produces every type at side $KM$ from every type at side $M$ for arbitrary $K$; and the resulting closed index function, which is the recursive form of the floor-function arrays of \\cite{Blake2017}.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24408",
  "author_total": 1,
  "authors": [
    {
      "name": "Sam Blake"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/659745w8h6mwj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24408",
  "fulltext_url": null,
  "id": "659745w8h6mwj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24408",
  "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": "W7223051895",
  "pdf_url": "https://arxiv.org/pdf/2610.24408",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/659745w8h6mwj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24408",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24408"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Recursive Construction of Perfect $K^n\\times K^n$ Arrays over $K$ Roots of Unity",
  "title_text": "A Recursive Construction of Perfect K^n× K^n Arrays over K Roots of Unity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/659745w8h6mwj",
  "venue": "arXiv",
  "versions": []
}