单项式表示的中心化子代数中的复广义加权矩阵
Complex generalised weighing matrices in centraliser algebras of monomial representations
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中文总结 AI 辅助
研究复广义加权矩阵\(CGW(n,w;k)\)，利用有限群单项式表示的中心化子代数，通过对舒尔覆盖线性特征的穷举搜索，对特定条件的此类矩阵分类，恢复相关无限族，构造量子纠错码并确定其最小距离。
中文摘要 AI 辅助
一个\(n×n\)矩阵\(W\)，每行每列恰有\(w\)个非零元素取自\(k\)次单位根集合，且满足\(WW^{\ast}=wI_n\)，则称其为复广义加权矩阵\(CGW(n,w;k)\)。我们通过有限群的单项式表示的中心化子代数来研究此类矩阵。利用对舒尔覆盖的线性特征的穷举搜索，我们对系数阶\(k\leq6\)且允许秩至多为五、次数至多为\(80\)的本原群通过强自同构作用的复广义加权矩阵进行了单项式等价分类，对于次数为\(100\)的情况有部分结果。普查恢复了与射影和仿射有限几何相关的已知无限族，描述了与汉明方案相关的无限族，并解决了文献中列举的一些小开放案例的存在性问题。我们从这些矩阵构造量子纠错码并精确确定其在所有情况下的最小距离。
英文摘要
An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.
