Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 44 – 52. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-52-3-44-52
Research Article
Full text in English

MSC 94B65, 52C17, 52C35, 46L08

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Noncommutative Spherical Codes

K. M. Krishna^{\ast}

School of Mathematics and Natural Sciences, Chanakya University Global Campus, NH-648, Haraluru
Village, Devanahalli Taluk, Bengaluru North District, Karnataka State, 562 110, India

Abstract. Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and information theory, continue to challenge researchers with their intricate structure and unresolved questions. Inspired by Polya’s heuristic principle of “vary the problem,” we extend the classical framework by introducing the notion of noncommutative spherical codes, with particular emphasis on the noncommutative Newton–Gregory kissing number problem. This generalization moves beyond the traditional Euclidean setting into the realm of operator algebras and Hilbert C*-modules, thereby opening new avenues of investigation. A cornerstone in the study of spherical codes is the celebrated Delsarte–Goethals–Seidel–Kabatianskii–Levenshtein linear programming bound, developed over the past half-century. This bound employs Gegenbauer polynomials to establish sharp upper limits on the size of spherical codes, and it has served as a fundamental tool in coding theory and discrete geometry. Remarkably, a recent elegant one-line proof by Pfender [J. Combin. Theory Ser. A, 2007] provides a streamlined derivation of a variant of this bound. We demonstrate that Pfender’s argument can be extended naturally to the setting of Hilbert C*-modules, thereby enriching the theory with noncommutative analogues.

Key words: Spherical code, Kissing number, Linear programming, Hilbert C*-module.

Received: 08.11.2025; Revised: 10.11.2025; Accepted: 18.11.2025; First online: 19.11.2025

For citation. Krishna K. M. Noncommutative spherical codes. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 44-52. EDN: FQZUDX. https://doi.org/10.26117/2079-6641-2025-52-3-44-52.

Funding. The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: kmaheshak@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Krishna K. M., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

  1. Zong C. Sphere packings, Universitext. New York: Springer-Verlag, 1999.
  2. Anstreicher K. M. The thirteen spheres: a new proof, Discrete Comput. Geom., 2004. vol. 31, no. 4,
    pp. 613–625.
  3. Bachoc C., Vallentin F. New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc., 2008. vol. 21, no. 3, pp. 909–924.
  4. B¨or¨oczky K. The Newton-Gregory problem revisited, In Discrete geometry, 2003. vol. 253, pp. 103–110.
  5. Boyvalenkov P., Dodunekov S., Musin O.A survey on the kissing numbers, Serdica Math. J., 2012.
    vol. 38, no. 4, pp. 507–522.
  6. Casselman B. The difficulties of kissing in three dimensions, Notices Amer. Math. Soc., 2004. vol. 51, no. 8, pp. 884–885.
  7. Glazyrin A.A short solution of the kissing number problem in dimension three, Discrete Comput.
    Geom., 2023. vol. 69, no. 3, pp. 931–935.
  8. Jenssen M., Joos F., Perkins W.On kissing numbers and spherical codes in high dimensions, Adv.
    Math., 2018. vol. 335, pp. 307–321.
  9. Kallal K., Kan T., Wang E. Improved lower bounds for kissing numbers in dimensions 25 through
    31., SIAM J. Discrete Math., 2017. vol. 31, no. 3, pp. 1895–1908.
  10. Kuklin N. A. Delsarte method in the problem on kissing numbers in high-dimensional spaces, Proc.
    Steklov Inst. Math., 2014. vol. 284, no. 1, pp. S108–S123.
  11. Leech J.The problem of the thirteen spheres, Math. Gaz., 1956. no. 40, pp. 22–23.
  12. Liberti L. Mathematical programming bounds for kissing numbers / Optimization and decision
    science: methodologies and applications, Springer Proc. Math. Stat., vol. 217. Cham, Springer, 2017, pp. 213–222.
  13. Machado F. C., de Oliveira Filho F. M. Improving the semidefinite programming bound for the kissing
    number by exploiting polynomial symmetry, Exp. Math., 2018. vol. 27, no. 3, pp. 362–369.
  14. Maehara H. The problem of thirteen spheres-a proof for undergraduates, European J. Combin., 2007.
    vol. 28, no. 6, pp. 1770–1778.
  15. Mittelmann H. D., Vallentin F. High-accuracy semidefinite programming bounds for kissing numbers,
    Experiment. Math., 2010. vol. 19, no. 2, pp. 175–179.
  16. Musin O. R. The kissing problem in three dimensions, Discrete Comput. Geom., 2006. vol. 35, no. 3,
    pp. 375–384.
  17. Musin O. R. The kissing number in four dimensions, Ann. of Math., 2008. vol. 168, no. 1, pp. 1–32.
  18. Odlyzko A. M., Sloane N. J. A. New bounds on the number of unit spheres that can touch a unit
    sphere in n dimensions, J. Combin. Theory Ser. A, 1979. vol. 26, no. 2, pp. 210–214.
  19. Pfender F. Improved Delsarte bounds for spherical codes in small dimensions, J. Combin. Theory
    Ser. A, 2007. vol. 114, no. 6, pp. 1133–1147.
  20. Pfender F., Ziegler G. M. Kissing numbers, sphere packings, and some unexpected proofs, Notices
    Amer. Math. Soc., 2004. vol. 51, no. 8, pp. 873–883.
  21. Sch¨utte K., van der Waerden B. L. Das Problem der dreizehn Kugeln, Math. Ann., 1953. no. 125,
    pp. 325–334.
  22. Bachoc C., Vallentin F. Semidefinite programming, multivariate orthogonal polynomials, and codes
    in spherical caps, European J. Combin., 2009. vol. 30, no. 3, pp. 625–637.
  23. Bannai E.,Etsuko Bannai E.A survey on spherical designs and algebraic combinatorics on spheres,
    European J. Combin., 2009. vol. 30, no. 6, pp. 1392–1425.
  24. Bannai E., Sloane N. J. A. Uniqueness of certain spherical codes, Canadian J. Math., 1981. vol. 33,
    no. 2, pp. 437–449.
  25. Barg A., Musin O. R. Codes in spherical caps, Adv. Math. Commun., 2007. vol. 1, no. 1, pp. 131–149.
  26. B¨or¨oczky K. J., Glazyrin A. Stability of optimal spherical codes, Monatsh. Math., 2024. vol. 205,
    no. 3, pp. 455–475.
  27. Boyvalenkov P. G., Dragnev P. D., Hardin D.P., Saff E. B., Stoyanova M. M. Universal lower bounds
    for potential energy of spherical codes, Constr. Approx., 2016. vol. 44, no. 3, pp. 385–415.
  28. Boyvalenkov P. G., Dragnev P. D., Hardin D.P., Saff E. B., Stoyanova M. M. Bounds for spherical
    codes: the Levenshtein framework lifted, Math. Comp., 2021. vol. 90, no. 329, pp. 1323–1356.
  29. Boyvalenkov P., Danev D. On maximal codes in polynomial metric spaces / Applied algebra,
    algebraic algorithms and error-correcting codes, Lecture Notes in Comput. Sci., vol. 1255. Berlin,
    Springer, 1997, pp. 29–38.
  30. Boyvalenkov P., Danev D., Landgev I.On maximal spherical codes. II, J. Combin. Des., 1999. vol. 7,
    no. 5, pp. 316–326.
  31. Cohn H., Jiao Y., Kumar A., Torquato S. Rigidity of spherical codes, Geom. Topol., 2011. vol. 15,
    no. 4, pp. 2235–2273.
  32. Conway J. H., Sloane N. J. A. Sphere packings, lattices and groups. New York: Springer-Verlag,
    1999.
  33. Delsarte P., Goethals J. M., Seidel J. J. Spherical codes and designs, Geometriae Dedicata, 1977. vol. 6,
    no. 3, pp. 363–388.
  34. Ericson T., Zinoviev V. Codes on Euclidean spheres, North-Holland Mathematical Library, vol. 63.
    Amsterdam: North-Holland Publishing Co., 2001.
  35. Musin O. R. Bounds for codes by semidefinite programming, Tr. Mat. Inst. Steklova, 2008. no. 263,
    pp. 143–158.
  36. Samorodnitsky A.On linear programming bounds for spherical codes and designs, Discrete Comput.
    Geom., 2004. vol. 31, no. 3, pp. 385–394.
  37. Sardari N. T., Zargar M. New upper bounds for spherical codes and packings, Math. Ann., 2024.
    vol. 389, no. 4, pp. 3653–3703.
  38. Sloane N. J. A.Tables of sphere packings and spherical codes, IEEE Trans. Inform. Theory, 1981.
    vol. 27, no. 3, pp. 327–338.
  39. Cohn H., Zhao Y. Sphere packing bounds via spherical codes, Duke Math. J., 2014. vol. 163, no. 10,
    pp. 1965–2002.
  40. Wegge-Olsen N. E. K-theory and C*-algebras: a friendly approach. Oxford: Oxford University
    Press, 1993.

Information about the author

Krishna Mahesh – PhD (Phys. Math.), Assistant Professor, School of Mathematics and Natural Sciences, Chanakya University Global Campus, Haraluru, India, ORCID 0000-0003-4872-8634.

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