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

It was previously shown by the authors that the discrete potentials of almost all known sharp codes attain universal lower bounds for polarization (PULB) for spherical $\tau$-designs, where "universal" is meant in the sense of applying to a large class of potentials that includes absolutely monotone functions of inner products and in the sense that the computational parameters of the bound are invariant with respect to the potential.

In this talk we characterize the sets of universal minima $D$ for some of these sharp codes $C$ found in the Leech lattice and establish a duality relationship, namely that the normalized discrete potentials $1$ of $C$ and $D$ have the same minimum value and the sets $C$ and $D$ are each others minima sets (up to antipodal symmetrization in some cases). The extremal duality is obtained by utilizing the natural embedding of the PULB pair codes in the Leech lattice and its properties, which simplifies the analysis significantly.

In the process we discover a new universally optimal code in the projective space $\mathbb{RP}^{21}$ with cardinality $1408$. Joint work with S. Borodachov (Towson University), P. Dragnev (Purdue University Fort Wayne, USA), D. Hardin and E. Saff (Vanderbilt University, Nashville, USA), M. Stoyanova (Sofia University, Bulgaria).