Sharpness of the intermediate orthoplex-bound leading-coefficient upper bound

Prove that, for every fixed dimension n≥4 and every integer k satisfying 3≤k≤n−1, the optimal high-SNR leading coefficient for SNR-wise optimized AWGN spherical codes with n+k codewords in dimension n equals K^{\ast}_{n+k,n}=4k(k−1).

Background

In the orthoplex-bound range, the paper studies spherical codebooks with M=n+k unit-norm codewords in dimension n, where 2≤k≤n. The best fixed packing-optimal codebook has leading coefficient K^{\mathrm{fix}}_{n+k,n}=4n(k−1). For every nonterminal value 2≤k<n, the authors construct an SNR-dependent family whose Gaussian soft-packing energy and exact ML error have leading coefficient 4k(k−1), thereby proving the upper bound K^{\ast}_{n+k,n}≤4k(k−1).

The equality is proved at the left endpoint k=2, where K^{\ast}_{n+2,n}=8, and at the terminal endpoint k=n, where K^{\ast}_{2n,n}=4n(n−1). For the intermediate range 3≤k≤n−1, the paper does not establish a matching global lower bound. In particular, it cannot exclude another jointly feasible multiscale deformation that produces a smaller value in the variational characterization of K^{\ast}_{n+k,n}. The conjecture asserts that the explicit moving-family upper bound is nevertheless sharp throughout this intermediate range.