A power Hadamard matrix $H(x)$ is a square matrix of dimension $n$ with entries from Laurent polynomial ring $L= \mathbb{Q} [x,x^{-1}]$ such that $H(x)H(x^{-1})^T=nI \mod f(x)$, where $f$ is some Laurent polynomial of degree greater than $0$. In the first part of this work, some new results on power Hadamard matrices are studied, where we mainly entend the work of Craigen and Woodford. In the second part, codes obtained from Butson-Hadamard matrices are discussed 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-homegeneous weight.

中文翻译：

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Power Hadamard 矩阵和 Plotkin 最优 $p^k$-ary 代码

幂 Hadamard 矩阵 $H(x)$ 是一个维度为 $n$ 的方阵，其条目来自 Laurent 多项式环 $L= \mathbb{Q} [x,x^{-1}]$ 使得 $H(x )H(x^{-1})^T=nI \mod f(x)$，其中 $f$ 是某个阶数大于 $0$ 的洛朗多项式。本工作的第一部分研究了幂Hadamard矩阵的一些新结果，其中主要涉及Craigen和Woodford的工作。第二部分讨论了从 Butson-Hadamard 矩阵获得的代码，并证明了这些代码的最小距离的一些界限。特别是，我们展示了从 Butson-Hadamard 矩阵获得的代码在非齐次权重下满足 Plotkin 界限。