Conjecture on universal polar dual pairs of spherical codes

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Let CC be a PULB-optimal code, and let DD be its set of universal minima. Assume that DD is in general position, meaning that DD is not contained in a hyperplane. Universal polar dual pair conjecture. The code CC together with DD forms a PULB-optimal pair (C,D)(C,D) and generates a maximal PULB-optimal pair, and hence a universal polar dual pair. This conjecture is motivated by the fact that all examples of PULB-optimal configurations found so far satisfy the asserted property; its general validity remains open.

References

Primary source

S. V. Borodachov, P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff and M. M. Stoyanova, “Universal polar dual pairs of spherical codes found in E_8 and Λ_24”, arXiv:2512.25037 (2025).

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