Two Quantum Latin Squares of Order 6 with Cardinalities 13 and 17

Abstract: We give two explicit quantum Latin squares of order $6$, one with cardinality $13$ and one with cardinality $17$. Throughout, vectors differing only by a global phase are counted as identical. The first construction is based on an orthogonal direct-sum decomposition $\mathbb{C}^6 = \mathbb{C}^4 \oplus \mathbb{C}^2$. The second construction is based on two-dimensional Hadamard pairs supported on coordinate planes.