Workshop

Spherical Codes and Designs

in the frame of the International Conference “Mathematics Days in Sofia – 2026”

Spherical codes and spherical designs play a central role in discrete geometry, coding theory, and numerical integration on the sphere. Recent advances have revealed deep connections with optimization, harmonic analysis, and applications ranging from communication systems to quantum information. A dedicated mini-symposium  would bring together researchers from these areas to exchange new results, identify emerging directions, and foster collaborations across disciplines.

The topics include, but are not limited to:

  1. Bounds for spherical codes and designs – linear programming and SDP
  2. Energy minimization problems
  3. Sphere packings and lattices
  4. Polarization problems
  5. Constructions of good spherical codes and designs
  6. Related topics – quantum spherical codes, etc.

Organizing Committee

  • Peter Boyvalenkov, Institute of Mathematics and Informatics – BAS, Bulgaria

  • Maya Stoyanova, Sofia University “St. Kliment Ohridski”, Bulgaria

  • Peter Dragnev, Purdue University Fort Wayne, USA

Participants

  • Alexey Glazyrin, The University of Texas Rio Grande Valley, USA

  • Damir Ferizovic, KU Leuven, Belgium

  • Danila Cherkashin, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria

  • Dmytro Bilyk, University of Minessota, USA

  • Ferenc Szollosi, Shimane University, Japan

  • Hiroshi Nozaki, Aichi University of Education, Japan

  • Marc Christian Zimmermann, University of Cologne, Germany

  • Martin Ehler, University of Vienna, Austria

  • Maya Stoyanova, Sofia University “St. Kliment Ohridski”, Bulgaria

  • Naser Talebizadeh Sardari, Penn State University, USA

  • Oleg Musin, University of Texas Rio Grande Valley, USA

  • Oleg Pikhurko, University of Warwick, UK

  • Peter Boyvalenkov, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bulgaria

  • Peter Dragnev, Purdue University Fort Wayne, USA

  • Ryutaro Misawa, Tohoku University, Japan

Program and Abstracts

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We study T-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate
Q-polynomial structure. Zhu (2021) provided a combinatorial characterization of T-designs in this scheme for certain index sets T, using a relationship between T-designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this talk, we present a characterization that applies to a strictly larger class of index sets T, based on a methodological extension of Delsarte’s original framework. This characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce (r,s)-designs, a new family of combinatorial objects arising from our characterization. We also discuss absolute lower bounds on the cardinality of (r,s)-designs and present several examples of tight designs attaining the bounds. Finally, we discuss related linear programming bounds and related phenomena for spherical codes.

This talk is based on joint work with Yuta Watanabe.

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We improve the previously best-known upper bounds on the sizes of θ-spherical codes for every θ < θ∗ ≈ 62.997◦ at least by a factor of 0.4325, in sufficiently high dimensions. Novelties of this paper include the analysis of triple correlations, usage of the concentration of mass in high dimensions, and the study of the spacings between the roots of Jacobi polynomials.

This is a joint work with Masoud Zargar.

In this talk will be presented an extension of known semidefinite and linear programming upper bounds for spherical codes. The idea is to use SDP (semidefinite programming) in addition to Gegenbauer polynomials to obtain new functions defining bounds for the distance distribution. I will also consider partitions of spherical codes by orthogonal hyperplanes for which this method yields new relations.

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The universal maxima on the sphere for discrete potentials induced by the sharp codes is investigated. Universality here is meant in the sense that the points attaining this absolute maximum over the unit sphere are the same for the class of absolutely monotone interaction potentials. It turns out that these maximizers are the points of the code itself. Moreover, we show that such codes are optimal for polarization (minimizing the absolute maximum value among all spherical codes of fixed cardinality and dimension) even for a larger class of potentials. This result was already established by Borodachov in 2022 for tight designs. For sharp spherical codes that are not tight designs, the notion of is introduced. Furthermore, this technique is applied to antipodal quasi-sharp codes. The current investigation complements recent work by the authors on universal polar dual pairs of spherical codes.

Joint work with S. Borodachov (Towson), P. Boyvalenkov (IMI-BAS), D. Hardin (Vanderbilt), E. Saff (Vanderbilt), M. Stoyanova (Sofia).

Spherical designs, introduced by Delsarte, Goethals, and Seidel in 1977, are important objects that have been studied extensively from both pure and applied mathematical viewpoints. Among the many problems surrounding spherical designs, this talk focuses in particular on the construction problem.

There is a substantial body of previous work on constructing spherical t-designs. However, if one seeks configurations with a small number of points, the pairs of dimensions and strengths t for which explicit constructions are available tend to be quite limited. On the other hand, if one aims for a general construction method that applies uniformly to arbitrary dimensions and strengths t, the resulting designs usually have a very large number of points.

In this talk, from the viewpoint of reconciling these two competing goals—small cardinality and general applicability—we introduce a new construction method for spherical designs using tight t-fusion frames. We also explain that this method provides a unified construction of several known beautiful spherical designs, and in particular, we present an explicit construction of spherical 5-designs.

The Event is Supported by: