We investigate single qudit and multiple qudit Pauli groups [1] and the quantum states/rays arising from their maximal bases.Remarkably,for the multiple qubit Pauli group Pd,d = 2n,the real rays are carried by a Barnes-Wall lattice Bn.We focus on the smallest subsets of rays allowing a proof of the Bell-Kochen-Specker theorem (BKS).BKS theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases.We investigate the geometrical structure of small BKS-proofs v-l involving v rays and l 2n-dimensional bases of n-qubits (1 < n < 5) [2].Specifically,we look at the parity proofs 18 - 9 with two qubits (A.Cabello,1996 [3]),the parity proofs 36–11 with three qubits (M.Kernaghan & A.Peres,1995 [4]) and a newly discovered nonparity proof 80-21 with four qubits (that improves a work at P.K Aravind’s group in 2008 [5]).One finds universal signatures of the distances among the bases,that carry various symmetries in their graphs.