Exact optimality of k-closest encoding for rotationally symmetric codebooks

Let $U^M=(U_1,\ldots,U_M)$ be a rotationally symmetric codebook, and let $k$-closest encoding denote the proposed encoding scheme that selects the $k$ closest codewords to the input. The $k$-closest encoding conjecture. The proposed $k$-closest encoding is exact-optimal for any rotationally symmetric codebook. The paper establishes exact optimality for the rotating simplex codebook and notes that the general claim would follow from the stated error formulation; whether $k$ may depend on the realization of $U^M$ remains unclear.