Let X⊂Rm be a spherical code (i.e., a finite subset of the unit sphere) and consider the ideal of all polynomials in m variables which vanish on X. Motivated by a study of cometric (QQ-polynomial) association schemes and spherical designs, we wish to determine certain properties of this ideal. After presenting some background material and preliminary results, we consider the case where X is the set of shortest vectors of one of the exceptional lattices E6, E7, E8, Λ24 (the Leech lattice) and determine for each: (i) the smallest degree of a non-trivial polynomial in the ideal and (ii) the smallest k for which the ideal admits a generating set of polynomials all of degree k or less. As it turns out, in all four cases mentioned above, these two values coincide, as they also do for the icosahedron, our introductory example. The paper concludes with a discussion of these two parameters, two open problems regarding their equality and a few remarks concerning connections to cometric association schemes.
Martin, W., & Steele, C. (2015). On the ideal of the shortest vectors in the Leech lattice and other lattices. Journal of Algebraic Combinatorics, 41(3), 707–726. https://doi.org/10.1007/s10801-014-0550-5
*denotes a WPI undergraduate student author