Three-point bounds proving optimality of Kerdock spherical and binary codes

Prove that for every integer k ≥ 2, the Kerdock spherical code in dimension 2^{2k}—consisting of 2^{4k} + 2^{2k+1} points with maximal inner product 1/2^{k} formed from 2^{2k-1}+1 real mutually unbiased bases—is optimal among all spherical codes, with the Bachoc–Vallentin three-point bounds certifying equality, and deduce optimality of the corresponding Kerdock binary codes of block length 2^{2k}.

Background

Kerdock codes produce large spherical codes with highly structured geometry (mutually unbiased bases) and are known to be optimal among antipodal codes. The paper proves optimality (and in some dimensions uniqueness) for k = 2, 3, 4, 5 via three-point bounds, and conjectures the method extends to all k ≥ 2.

Establishing this would also settle optimality of the associated Kerdock binary codes at all these lengths and show that Schrijver’s three-point bound is sharp in these cases.