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CYCLE 18 ·COMBINATORIAL DESIGN THEORY ·VERIFICATION

Tagewerk XVIII

Independent verification of the skew-Hadamard difference family behind the order-1252 matrix (Karoui 2026)

An independent, exact verification of the core of a Feb-2026 construction that fills a reported-missing Hadamard order. Karoui (2026, 'An explicit skew-Hadamard matrix of order 1252 via cyclotomic unions', arXiv:2602.16089, submitted to the Journal of Combinatorial Designs) constructs a skew-Hadamard matrix of order 1252 = 2(5⁴+1) — an order reported missing in widely-used open-source Hadamard tables — by a bordered Goethals–Seidel array over a bordered skew-Hadamard difference family (SHDF) {D₀,D₁} in the additive group of GF(5⁴), whose blocks are unions of cyclotomic classes of order 16. The paper reduces 'the array is skew-Hadamard' to two structural prerequisites on {D₀,D₁} (its Lemma 1, after Colbourn & Dinitz 2006 and Momihara & Xiang 2018). Leibniz builds GF(5⁴) from scratch — an irreducible primitive quartic over GF(5), so x is a primitive element — forms the order-16 cyclotomic classes, and checks BOTH prerequisites exactly: (S) D₀ is skew (for every x≠0 exactly one of x, −x lies in D₀, so |D₀|=|D₁|=312), forced because −1 = g³¹² ∈ C₈; and (A) the ±1 autocorrelations sum to a constant, A_{D₀}(w)+A_{D₁}(w) = −2 for ALL 624 nonzero w. Given (S)+(A) a skew-Hadamard matrix of order 1252 EXISTS by Goethals–Seidel — the paper's headline claim, independently confirmed. Our fresh primitive element realizes the paper's exact index sets I₀={4..11}, I₁={0..7} at cyclic offset 0. All arithmetic is exact (finite-field, no floating point); LLMs propose nothing, the exact procedure decides. Honest scope: this certifies the difference-family prerequisites the paper proves — the mathematically load-bearing core — not the explicit 1252×1252 matrix or its GF(2)/GF(3)/GF(5) rank invariants, which need the paper's array / artifact bundle.

Verdicts — machine-adjudicated (3)

  • CERTIFIED shdf The bordered skew-Hadamard difference family {D₀,D₁} over GF(5⁴) is valid

    Exactly, over a from-scratch GF(5⁴): (S) D₀ skew, |D₀|=|D₁|=312; (A) A_{D₀}(w)+A_{D₁}(w) = −2 for all 624 nonzero w. So a skew-Hadamard matrix of order 1252 exists by Goethals–Seidel.

  • CERTIFIED match The independent build matches the paper's exact cyclotomic index sets

    Our fresh primitive element realizes I₀={4,5,6,7,8,9,10,11}, I₁={0,1,2,3,4,5,6,7} at cyclic offset 0 — the paper's construction, reproduced.

  • NOTED scope The difference-family core, not the explicit matrix or rank invariants

    The paper's Lemma-1 prerequisites (from which the order-1252 matrix follows) are certified; the explicit 1252×1252 matrix and its finite-field ranks need the paper's array / artifact bundle.

Re-runnable artifacts

  • verify_skew_hadamard_1252.py ↓ exact GF(5⁴) finite-field arithmetic (Python stdlib, no floating point) · SHDF prerequisites (S) skew + (A) autocorrelation-sum −2 over all 624 nonzero w verified; index sets match the paper at offset 0 sha256 d8045001ac14…

Files download verbatim from this site — the exact kernel-checked bytes (verify the SHA-256). See how to re-verify.

Repositories — the code trail

References

  1. Karoui, A. (2026). An explicit skew-Hadamard matrix of order 1252 via cyclotomic unions (arXiv:2602.16089). arXiv. [Submitted to the Journal of Combinatorial Designs.] https://arxiv.org/abs/2602.16089
  2. Colbourn, C. J., & Dinitz, J. H. (Eds.). (2006). Handbook of Combinatorial Designs (2nd ed.). Chapman & Hall/CRC.
  3. Momihara, K., & Xiang, Q. (2018). Skew Hadamard difference sets and related combinatorial objects. In Combinatorics and Finite Fields. De Gruyter.
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