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Cyclotomic Constraints and Cocyclic Structures for Turyn–Golay Quadruples

Ilias Kotsireas, Tanush Shaska

I-AIMS preprint sh-171 · 2026

A Turyn–Golay quadruple of length n is a quadruple of {−1, +1} sequences whose non-periodic autocorrelations sum to zero at every nonzero shift; existence for every n implies the Hadamard conjecture. In the companion paper we showed that the defining conditions reduce to group-ring identities over the finite cyclic rings ℤ[ℤₘ], m ≤ M(n), with M(167) = 33. This paper develops the arithmetic of these reductions. We prove four theorems on the character values h_Q(ζₘ) of the member sequences: a nonvanishing lemma, a two-unit theorem at the self-conjugate moduli, an inert coprimality theorem, and a norm-bound criterion p²ᶠ > (4n)ᵠ⁽ᵐ⁾ requiring no self-conjugacy hypothesis, whose reach at length 167 is provably exactly the sixteen inert moduli. At the moduli dividing 168 the two-unit theorem becomes a set of explicit rational congruence filters. We then enumerate admissible four-member character profiles at levels m ≤ 3 and the complete achievable single-sequence character data through level m = 7: at level 7 — the first split self-conjugate modulus — exactly 77 achievable values of |h_Q(ζ₇)|² are divisible by each of the three primes above 167 and none by two of them, so each member sequence whose character norm is divisible by a prime above 167 must lie in one of three explicit 77-element sets of character norm values. The existence of TG(167) remains open.

MSC classes: Primary 05B20, 05B10, 11R18; Secondary 20J06, 94A55

References

  1. Jean-Marie Goethals and Johan J. Seidel, Orthogonal matrices with zero diagonal, Canad. J. Math. 19 (1967), 1001–1010.
  2. Hadi Kharaghani and Behruz Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combin. Des. 13 (2005), 435–440.
  3. Shalom Eliahou, A 64-modular Hadamard matrix of order 668, Australas. J. Combin. 93 (2025), 422–427.
  4. Levent Alpöge et al., Hadamard matrices of order 668 and of all orders below 2000, 2026. Announcement, August 2026.
  5. Ilias Kotsireas and Tony Shaska, Group-theoretic approach to binary sequences and correlations, I: the aperiodic theory, 2026. In preparation.
  6. Richard J. Turyn and James Storer, On binary sequences, Proc. Amer. Math. Soc. 12 (1961), 394–399.
  7. Richard J. Turyn, Character sums and difference sets, Pacific J. Math. 15 (1965), 319–346.
  8. Bernhard Schmidt, Characters and cyclotomic fields in finite geometry, Lecture Notes in Mathematics, vol. 1797, Springer-Verlag, Berlin, 2002.
  9. Ka Hin Leung and Bernhard Schmidt, The field descent method, Des. Codes Cryptogr. 36 (2005), 171–188.
  10. Bernhard Schmidt, Cyclotomic integers and finite geometry, J. Amer. Math. Soc. 12 (1999), 929–952.
  11. Shalom Eliahou, Michel Kervaire, and Bahman Saffari, A new restriction on the lengths of Golay complementary sequences, J. Combin. Theory Ser. A 55 (1990), 49–59.
  12. ———, On Golay polynomial pairs, Adv. in Appl. Math. 12 (1991), 235–292.
  13. Kathy J. Horadam and Warwick de Launey, Cocyclic development of designs, J. Algebraic Combin. 2 (1993), 267–290.
  14. Padraig O Cath́áin and Marc Röder, The cocyclic Hadamard matrices of order less than 40, Des. Codes Cryptogr. 58 (2011), 73–88.
  15. Kathy J. Horadam, Hadamard matrices and their applications, Princeton University Press, Princeton, NJ, 2007.
  16. Warwick de Launey and Dane L. Flannery, Algebraic design theory, Mathematical Surveys and Monographs, vol. 175, American Mathematical Society, Providence, RI, 2011.
  17. Víctor Alvarez, Jośé Andrés Armario, María Dolores Frau, and Pedro Real, The homological reduction method for computing cocyclic Hadamard matrices, J. Symbolic Comput. 44 (2009), 558–570.
  18. Dane L. Flannery, Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra 192 (1997), 749–779.
  19. Noboru Ito, On Hadamard groups, J. Algebra 168 (1994), 981–987.
  20. Bernhard Schmidt, Williamson matrices and a conjecture of Ito’s, Des. Codes Cryptogr. 17 (1999), 61–68.
  21. Richard J. Turyn, Hadamard matrices, Baumert–Hall units, four-symbol sequences, pulse compression, and surface wave encodings, J. Combin. Theory Ser. A 16 (1974), 313–333.
  22. Padraig O Cath́áin, Cohomology, extensions and automorphisms of pairwise combinatorial designs, Master’s Thesis, 2011. Available at https://maths.nuigalway.ie/~padraig/Docs/ PadraigOCathainMastersThesis.pdf.
  23. Víctor Alvarez, Jośé Andrés Armario, Raúl M. Falcón, María Dolores Frau, Félix Gudiel, María Belén Güemes, and Amparo Osuna, Goethals–Seidel loops and LFSR based pseudococyclic Hadamard matrices, Mathematics 8 (2020), no. 1, 24.
  24. John Williamson, Hadamard’s determinant theorem and the sum of four squares, Duke Math. J. 11 (1944), 65–81.
  25. Marcel J. E. Golay, Complementary series, IRE Trans. Inform. Theory 7 (1961), 82–87.
  26. Chung H. Yang, On composition of four-symbol δ-codes and Hadamard matrices, Proc. Amer. Math. Soc. 107 (1989), 763–776.
  27. Jennifer Seberry Wallis, On the existence of Hadamard matrices, J. Combin. Theory Ser. A 21 (1976), 188–195.
  28. Robert Craigen, Signed groups, sequences, and the asymptotic existence of Hadamard matrices, J. Combin. Theory Ser. A 71 (1995), 241–254.
  29. Warwick de Launey and Daniel M. Gordon, A comment on the Hadamard conjecture, J. Combin. Theory Ser. A 95 (2001), 180–184. CARGO Lab, Wilfrid Laurier University, Waterloo, Ontario, Canada

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