{
  "abstract": "We propose Q(q,d), a family of q-ary non-CSS qudit stabilizer codes with prime power q and dimension d >= 2, constructed from Singer difference sets and non-degenerate quadrics in PG(d,q). The parity check H = (A | M_Q A) couples the Singer-circulant incidence matrix A with the quadric circulant M_Q, and a Cancellation Lemma uses the identity (k - lambda) = q^{d-1} = 0 mod p to make H symplectic-isotropic over F_q. Five structural theorems carry the novelty: a cross-correlation modular-constancy identity nu = 1 mod q at q odd, d >= 3, with a two-value pattern that proves a 2-torsion distance ceiling; a CRT zero-count identity; an explicit Clifford bridge identifying Q(q,2) as a graph-state encoding of the trivial Singer CSS code, with the 2021 low-density-spreading (LDS) sign-flip pattern realised by a symmetric circulant; a paired column-rigidity equality d_min = R_P + 1; and a lower bound d_min(Q(7,2)) >= 8 from an exhaustive Singer-Frobenius search. A Hamada-Smith specialisation yields the closed-form logical dimension k(p,d) = theta_d(p) - C(p+d-1, d) - 1, and the flagship Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p attains asymptotic rate one-half. We conjecture d_min(Q(p,2)) = p + 1 for odd primes, proven at p = 5 and lower-bounded to >= 8 at p = 7. Instances include Q(3,2) = [[13,6,3]]_3, Q(5,2) = [[31,15,6]]_5, Q(7,2) = [[57,28, d>=8]]_7, and Q(3,4) = [[121,105,3]]_3. Monte-Carlo over 1.5 x 10^6 trials shows Q(5,2) achieves a 70-fold per-logical-qudit improvement over Steane [[7,1,3]]_2 at depolarising rate 10^-3 with zero silent logical errors. A textbook bounded-distance decoder is used; Berlekamp-Massey-style cyclic decoding is left as the natural sequel.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02392",
  "author_total": 2,
  "authors": [
    {
      "name": "Michel Kulhandjian"
    },
    {
      "name": "Lajos Hanzo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cexmdxfkbav9c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02392",
  "fulltext_url": null,
  "id": "cexmdxfkbav9c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02392",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7220438206",
  "pdf_url": "https://arxiv.org/pdf/2610.02392",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cexmdxfkbav9c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2610.02392",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02392"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance",
  "title_text": "Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/cexmdxfkbav9c",
  "venue": "arXiv",
  "versions": []
}