{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R14",
  "content_hash": null,
  "slug": "apn256-attempt-classification-audit",
  "type": "attempt",
  "title": "Classification and computational-search audit",
  "summary": "Large restricted searches have produced strong exclusions, while each source records a boundary short of all 256-element-field permutations.",
  "relevance": "For An APN permutation of the 256-element field, record apn256-attempt-classification-audit (“Classification and computational-search audit”) documents a concrete method, search boundary, or failed route. The record states: Large restricted searches have produced strong exclusions, while each source records a boundary short of all 256-element-field permutations.",
  "relevance_source": "recorded",
  "body": "The audit followed three computational programs. The 2021 self-equivalence search treats permutations with a nontrivial linear self-equivalence after CCZ transformation. It exhausted all except a few dimension-eight automorphism classes and found no new APN permutation. This is a structural search rather than a classification of every permutation.\n\nThe 2025 quadratic search generated 3,775,599 inequivalent APN functions in eight variables and tested their CCZ classes for permutation representatives. No class passed. The authors estimate roughly six million quadratic APN classes in total, so their database is explicitly incomplete.\n\nThe June 2026 Gröbner-basis search covered 428 of 65,536 normalized slices around one order-five self-equivalence subspace. It found four classes absent from the 2025 database. The source states that its representatives are nonpermutations and leaves CCZ-equivalent permutation testing for future work. Together these searches rule out power permutations and several million quadratic CCZ classes. They do not cover arbitrary cubic or higher-degree permutations.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2606.11967",
      "locator": "Kuznetsov (2026), Sections I-C, VI-K, VII-B, VIII, and IX; Beierle et al. (2025), Introduction and Sections 4-5; Beierle, Brinkmann, and Leander (2021), abstract and search classification"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2606.11967",
    "locator": "Kuznetsov (2026), Sections I-C, VI-K, VII-B, VIII, and IX; Beierle et al. (2025), Introduction and Sections 4-5; Beierle, Brinkmann, and Leander (2021), abstract and search classification"
  },
  "models": [],
  "relations": [
    {
      "slug": "R16",
      "title": "Existence of an APN permutation on F_256 remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R13",
      "title": "Exact differential-uniformity verifier and power-permutation scan",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "apn-permutation-f256",
      "title": "apn permutation f256",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}