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An infinite family of Goethals-Seidel arrays

Algebra · math.CO · math.RT · posed by Mingyuan Xia, Tianbing Xia, Jennifer Seberry · open

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Statement

Do there exist polynomials f_{1}(\zeta),f_{2}(\zeta),f_{3}(\zeta) and f_{4}(\zeta) ,corresponding to every pair (a, b), satisfying (33), given as in (15), satisfying (16)and (17), such that f_{1}(1)=0,\quad f_{2}(1)^{2}=a^{2},\quad f_{3}(1)^{2}=f_{4}(1)^{2}=b^{2}, or not?

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  • An infinite family of Goethals-Seidel arrays
  • FAR
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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    NEW

    Problem: Reconstructed question: in Xia--Xia--Seberry’s construction, for every integer pair (a,b)(a,b) satisfying

    4m−1=a2+2b2,4m-1=a^{2}+2b^{2},

    do the polynomials f1,f2,f3,f4f_1,f_2,f_3,f_4 defined as in their formula (15), and satisfying their conditions (16),(17), exist with

    f1(1)=0,f2(1)2=a2,f3(1)2=f4(1)2=b2?f_1(1)=0,\qquad f_2(1)^2=a^2,\qquad f_3(1)^2=f_4(1)^2=b^2?

    Here formula (15) is the finite-field construction from a primitive element g∈GF(q2)g\in GF(q^2), q=4m−1q=4m-1, writing gk=αkx+βkg^k=\alpha_k x+\beta_k and using the quadratic character χ\chi of GF(q)GF(q).

    Result: The answer is no.

    Take q=27q=27. Then

    27=52+2⋅12=32+2⋅32,27=5^2+2\cdot1^2=3^2+2\cdot3^2,

    so the pair (a,b)=(3,3)(a,b)=(3,3) satisfies (33).

    Work in

    F=GF(27)=F3[u]/(u3−u+1),F=GF(27)=\mathbb F_3[u]/(u^3-u+1),

    and let γ=2u2\gamma=2u^2. Since γ13=−1\gamma^{13}=-1, γ\gamma is nonsquare, so E=F[x]/(x2−γ)≅GF(272)E=F[x]/(x^2-\gamma)\cong GF(27^2). Let

    g=u2x+u2.g=u^2x+u^2.

    A direct multiplication in EE gives

    g728=1,g364=−1,g104≠1,g56≠1,g^{728}=1,\quad g^{364}=-1,\quad g^{104}\ne1,\quad g^{56}\ne1,

    so gg is primitive, since 728=272−1=23⋅7⋅13728=27^2-1=2^3\cdot7\cdot13.

    Writing gk=αkx+βkg^k=\alpha_kx+\beta_k, and denoting the quadratic character of FF by χ\chi, the sums appearing in (15) are

    Sα(r)=∑i=06χ(αr+8i),Sβ(r)=∑i=06χ(βr+8i).S_\alpha(r)=\sum_{i=0}^{6}\chi(\alpha_{r+8i}),\qquad S_\beta(r)=\sum_{i=0}^{6}\chi(\beta_{r+8i}).

    Direct calculation gives

    (Sα(0),Sβ(0))=(0,−5),(S_\alpha(0),S_\beta(0))=(0,-5),

    and for the odd residue classes,

    (Sα(1),Sβ(1))=(1,−1),(S_\alpha(1),S_\beta(1))=(1,-1), (Sα(3),Sβ(3))=(−1,−1),(S_\alpha(3),S_\beta(3))=(-1,-1), (Sα(5),Sβ(5))=(−1,1),(S_\alpha(5),S_\beta(5))=(-1,1), (Sα(7),Sβ(7))=(1,1).(S_\alpha(7),S_\beta(7))=(1,1).

    Every other primitive element is gsg^s with gcd⁡(s,728)=1\gcd(s,728)=1. Since the character sequences have period 5656, replacing gg by gsg^s only permutes the class 8i8i and sends the class 8i+18i+1 to one of 1,3,5,7(mod8)1,3,5,7 \pmod 8. Replacing γ\gamma by another nonsquare rescales the xx-coordinate and changes only a global sign in the α\alpha-sums, hence not their squares.

    Thus every polynomial quadruple produced by (15) for q=27q=27 has

    f1(1)=0,f2(1)2=25,f3(1)2=f4(1)2=1,f_1(1)=0,\qquad f_2(1)^2=25,\qquad f_3(1)^2=f_4(1)^2=1,

    never

    f2(1)2=f3(1)2=f4(1)2=9.f_2(1)^2=f_3(1)^2=f_4(1)^2=9.

    Therefore the pair (a,b)=(3,3)(a,b)=(3,3) is not realized, disproving the “every pair” question.

    Citation: M. Xia, T. Xia, and J. Seberry, “An infinite family of Goethals-Seidel arrays,” Discrete Applied Mathematics 145 (2005), 498–504.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The claimed counterexample attacks the correct Remark 2 statement. For q=27q=27, the pair (a,b)=(3,3)(a,b)=(3,3) satisfies 27=a2+2b227=a^2+2b^2. The finite-field computation shows that the construction (15) yields squared values (f2(1)2,f3(1)2,f4(1)2)=(25,1,1)(f_2(1)^2,f_3(1)^2,f_4(1)^2)=(25,1,1), not (9,9,9)(9,9,9). The argument also accounts for all choices of primitive element and nonsquare parameter via the period 5656 and rescaling. I found no indication that this exact resolution is already known.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted result appears genuinely new, but it is a narrow finite computation: a single small counterexample q=27q=27, (a,b)=(3,3)(a,b)=(3,3), to a question about the specific Xia--Xia--Seberry--Wu construction. It does not resolve a broad Hadamard/Goethals--Seidel existence problem or introduce a general method. At most it would support a short note/corrigendum, not a standalone standard journal paper.

    Literature check: I searched the exact paper title/DOI, “Remark 2”, the conditions f1(1)=0f_1(1)=0, f2(1)2=a2f_2(1)^2=a^2, q=a2+2b2q=a^2+2b^2, “Xia Seberry Goethals-Seidel”, “GF(27)”, and “q=27” across OpenAlex metadata, arXiv full-text search, Internet Archive full-text search, OEIS, GitHub/forum-style searches, and accessible related Goethals--Seidel papers. I found the original article and later related work on Goethals--Seidel difference families, including Đoković’s 2024 arXiv paper listing cyclic GS families for v=27v=27, but not a prior resolution of this specific finite-field-polynomial question or the (q,a,b)=(27,3,3)(q,a,b)=(27,3,3) nonrealizability counterexample.

    Citation: M. Xia, T. Xia, J. Seberry, and J. Wu, “An infinite family of Goethals–Seidel arrays,” Discrete Applied Mathematics 145 (2005), 498–504. DOI: 10.1016/j.dam.2003.06.007.

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