阶数为六、基数为十九、二十一和二十三的量子拉丁方
New Cardinalities for Quantum Latin Squares of Order Six
AI总结：
研究给出阶数为6、基数分别为19、21和23的量子拉丁方。前两个源于复哈达玛矩阵列的归一化舒尔积，第三个基于直和构造。这些结果与之前构造一起确定了\(6\leq c\leq24\)区间内除\(c = 7\)外的所有基数。
AI中文摘要：
在仅相差一个全局相位的向量视为相同的情况下，我们给出了三个阶数为6、基数分别为19、21和23的显式量子拉丁方。前两个例子源于复哈达玛矩阵列的归一化舒尔积。对于基数19，一个八次单位根上的布特森型矩阵有唯一非平凡重合。对于基数21，卡尔森三参数族的一个显式成员有21个两两不等价的无序舒尔积。为超越对称舒尔积界，基于\(\C^6 = \C^4\oplus\C^2\)分解给出第三个直和构造，利用19条四维射线和4条二维射线，每行每列是正交基，得基数23。这些结果与之前的构造确定了\(6\leq c\leq24\)区间内除\(c = 7\)外的所有基数。
英文摘要：
We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, $27$, $32$, and $35$, where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$, these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$.
