{
  "abstract": "We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over $\\mathbb{F}_{41}$ or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over $\\mathbb{F}_{43}$; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, $n^2/18+n^{3/2}/360$, with an $O(n)$ remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28690",
  "author_total": 2,
  "authors": [
    {
      "name": "Bogdan Dumitru"
    },
    {
      "name": "Cristian Budala"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c55554x3yv805.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28690",
  "fulltext_url": null,
  "id": "c55554x3yv805",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28690",
  "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.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7204968019",
  "pdf_url": "https://arxiv.org/pdf/2608.28690",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c55554x3yv805.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28690",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28690"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42",
  "title_text": "An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/c55554x3yv805",
  "venue": "arXiv",
  "versions": []
}