A Butson-Hadamard matrix H is a square matrix of dimension n whose entries are complex roots of unity such that HH*= nI. In the first part of this work, some new results on generalized Gray map are studied. In the second part, codes obtained from Butson-Hadamard matrices and some bounds on the minimum distance of these codes are proved. In particular, we show that the code obtained from a Butson-Hadamard matrix meets the Plotkin bound under a non-homogeneous weight. We also give the parameters of some code families which are obtained from modified Butson-Hadamard matrices under a (non)homogeneous Gray map.

翻译：Butson-Hadamard 矩阵H是维度的平方矩阵,其条目具有复杂的统一根基,因此HH ⁇ nI。在这项工作的第一部分,研究了关于通用灰色地图的一些新结果。在第二部分,证实了从布森-Hadamard矩阵获得的代码和这些代码最低距离的一些界限。特别是,我们表明从布森-Hadamard矩阵获得的代码符合非均匀重量下约束的Plotkin。我们还给出了从一个(非均匀)混合灰色地图下修改的布森-哈达马德矩阵获得的一些代码家庭参数。