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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) in \zeta of order 6 , given as in (15), satisfying (16) and (17), such that f_{1}(1)=0,\quad f_{2}(1)^{2}=f_{3}(1)^{2}=f_{4}(1)^{2}=3^{2}?

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  • An infinite family of Goethals-Seidel arrays
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  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 statement: in Xia–Xia–Seberry–Wu’s formula (15), for order 66 (i.e. q=4⋅6+3=27q=4\cdot6+3=27), do there exist the four polynomials f1,…,f4f_1,\dots,f_4, satisfying (16),(17), with

    f1(1)=0,f2(1)2=f3(1)2=f4(1)2=32?f_1(1)=0,\qquad f_2(1)^2=f_3(1)^2=f_4(1)^2=3^2?

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

    Result: No.

    Let

    F=GF(27)=F3[u]/(u3−u+1),γ=2u2,F=GF(27)=\mathbb F_3[u]/(u^3-u+1),\qquad \gamma=2u^2,

    so γ13=−1\gamma^{13}=-1, hence γ\gamma is nonsquare. Put

    E=F[x]/(x2−γ)E=F[x]/(x^2-\gamma)

    and take

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

    A direct exact computation in EE gives

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

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

    Write gk=αkx+βkg^k=\alpha_kx+\beta_k. For the character sums occurring in (15),

    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}),

    one obtains

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

    and for odd r(mod8)r\pmod 8,

    r1357Sα(r)1−1−11Sβ(r)−1−111\begin{array}{c|cccc} r&1&3&5&7\\ \hline S_\alpha(r)&1&-1&-1&1\\ S_\beta(r)&-1&-1&1&1 \end{array}

    Thus formula (15) gives

    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,

    not 9,9,99,9,9.

    This is independent of the primitive element chosen: every primitive element is gsg^s with gcd⁡(s,728)=1\gcd(s,728)=1, and g56∈Fg^{56}\in F is a square, so the character sequences have period 5656; multiplication by ss merely permutes the relevant residue classes. Changing the nonsquare γ\gamma only rescales all αk\alpha_k by a fixed nonzero field element, affecting at most signs and not squares.

    Hence no polynomials of the required form exist.

    Citation: Source problem: M. Xia, T. Xia, J. Seberry, J. Wu, “An infinite family of Goethals–Seidel arrays,” Discrete Applied Mathematics 145 (2005), 498–504. No prior resolving citation used.

  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 disproof attacks the correct order-6 specialization of the Remark 2 question. The finite-field computation for q=27q=27 is internally consistent: it gives f1(1)=0f_1(1)=0 but (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 covers all primitive elements via the period 5656, and changes of nonsquare parameter only affect squared sums by harmless global signs. I found no evidence of a prior published resolution of this exact nonrealizability statement.

    Novelty assessment

    TYPE1

    Classification rationale: Genuinely new as far as I could determine, but very minor: it is a finite exact computation for the single order-6/q=27q=27 instance of a specialized Xia–Xia–Seberry–Wu construction. It does not give a general method or resolve a broad Goethals–Seidel/Hadamard problem. At most it supports a short note or corrigendum, not a standalone standard journal paper.

    Literature check: I searched the exact title/DOI, “Remark 2,” f1(1)=0f_1(1)=0, f2(1)2=f3(1)2=f4(1)2=32f_2(1)^2=f_3(1)^2=f_4(1)^2=3^2, GF(27)GF(27), q=27q=27, (a,b)=(3,3)(a,b)=(3,3), a2+2b2a^2+2b^2, and related Goethals–Seidel array/difference-family terminology in OpenAlex, arXiv, Internet Archive full text, OEIS, GitHub/forum-style sources, and related GS difference-family papers. I found related classification work, e.g. Đoković–Kotsireas on cyclic GS difference families for small odd orders, but not this finite-field-polynomial nonexistence computation or a stronger statement implying it.

    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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