Let G be a finite group and let h be a positive integer. A BH(G, h) matrix is a G-invariant ∣G∣ × ∣G∣ matrix H whose entries are complex hth roots of unity such that H H* = ∣G∣I_∣G∣, where H* denotes the complex conjugate transpose of H, and I_∣G∣ denotes the identity matrix of order ∣G∣. In this paper, we give three new constructions of BH(G, h) matrices. The first construction is the first known family of BH(G, h) matrices in which G does not need to be abelian. The second and the third constructions are two families of BH(G, h) matrices in which G is a finite local ring.

中文翻译：

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群不变Butson Hadamard矩阵的新构造

设G为有限群，h为正整数。BH( G, h ) 矩阵是G不变的 ∣ G ∣ × ∣ G ∣ 矩阵H，其条目是复数h th 个单位根，使得HH* = ∣ G ∣ I _{∣ G ∣}，其中H * 表示复数H 的共轭转置，I _{∣ G ∣}表示阶∣ G ∣的单位矩阵。在本文中，我们给出了 BH( G, h) 矩阵。第一个构造是第一个已知的 BH( G, h ) 矩阵族，其中G不需要是阿贝尔矩阵。第二个和第三个构造是两个 BH( G, h ) 矩阵族，其中G是有限局部环。