AI 中文总结
研究有限域上广义加权哈达玛矩阵，探讨其存在性、性质与构造，证明可逆矩阵子集构成群，引入等价概念及证明商群是阿贝尔群，并讨论其在编码理论中的应用。
AI 中文摘要
本文探讨了有限域上广义加权哈达玛（GWH）矩阵的存在性、若干性质及构造。研究了可逆GWH矩阵子集，证明其在矩阵乘法下构成群。引入通过正交变换定义的矩阵间强等价概念，证明正交矩阵子群对应的商群是阿贝尔群。最后讨论了这些矩阵在编码理论中的一些应用。
英文摘要
The existence, several properties, and constructions of Generalized Weighing-Hadamard (GWH) matrices over finite fields are addressed in this work. We study the subset of invertible GWH matrices and show that it forms a group under matrix multiplication. Besides that, we introduce a strong notion of equivalence between such matrices, defined via orthogonal transformations, and further prove that the corresponding quotient group by the subgroup of orthogonal matrices is abelian. Finally, we discuss some applications of these matrices in coding theory
