Spherical optimality of the 2816-point Leech-lattice code

Determine whether the 2816-point code A_2=(22,2816,5) is optimal among all spherical codes on S^{21} with 2816 points, rather than only among antipodal spherical codes.

Background

The code A_2=(22,2816,5) arises as a derived configuration from the Leech lattice and consists of 1408 antipodal pairs. The paper proves that the corresponding 1408-line projective code is universally optimal in RP{21}, and consequently that A_2 is universally optimal among antipodal spherical codes of cardinality 2816.

The authors emphasize that this does not establish optimality among arbitrary, not necessarily antipodal, spherical codes. The standard Levenshtein bound is substantially weaker than the required cardinality threshold, and the paper notes that linear programming cannot improve that bound in the relevant setting.

References

The problem of optimality of $A_2$ as a spherical code is open.

— Universal polar dual pairs of spherical codes found in $E_8$ and $Λ_{24}$  (2512.25037 - Borodachov et al., 31 Dec 2025) in Section 6, A new universally optimal code in the real projective space RP^{21}