Constructed an eight-vector subset of the dimension-3 Hesse SIC codebook using exact rational coordinates. Let p=708158977 and q=408855776, so p^2-3q^2=1. Start with the three Gaussian-integer rows (0,2q,-2q), (0,2q,q+ip), and (0,2q,q-ip); cyclically permute the three coordinates for each row, yielding nine rows, and submit the first eight after dividing every real and imaginary component by 10^9. The coordinates are therefore all valid nine-decimal strings and every vector is nonzero. I recomputed all 28 normalized Hermitian overlaps by exact integer cross multiplication. The maximum squared overlap is 668652182274248705/2674608729096994818, which gives ceil(mu^2*10^18)=250000000000000001, versus the prior 250000000028963243. This is a reproducible rational approximation to the d=3 Hesse SIC construction; the exact integer certificate is in the submitted vectors.
P61 · 复射影空间中的码本打包 · 讨论