论文标题
Butson完整的推出代码
Butson full propelinear codes
论文作者
论文摘要
在本文中，我们研究了Butson Hadamard矩阵，以及以对数形式的这些矩阵（称为BH-codes）的有限环的编码。我们通过对数形式的矩阵介绍了Butson Hadamard矩阵的新形态，这与最近关于ÓCatháin和Swartz的词语中给出的形态相媲美。也就是说，我们表明，如果给出了$ k^{\ rm th} $ Unity的Butson Hadamard矩阵，我们可以在$ \ ell^{\ rm th} $ unity上构建一个更大的Butson Matrix，以提供任何$ \ ell $ k $ k $的$ p $ p $ k $ k $ gul divide $ ell undies $ k $ g。 我们证明了$ \ mathbb {z} _ {p^s} $ - 带有$ p $的添加代码，质量数字是同构的，作为$ \ m \ m mathbb {z} _ {p^s} $的bh-code的组，该bh代码在灰色地图下的图像，$ bh code是$ \ math code of $ \ math BBB { $ P = 2 $）。此外，我们研究了当Butson矩阵是Cocyclic时，我们研究了这些代码（及其图像）的固有的推进结构。研究了这些代码的某些结构特性，并提供了示例。
In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of Ó Catháin and Swartz. That is, we show how, if given a Butson Hadamard matrix over the $k^{\rm th}$ roots of unity, we can construct a larger Butson matrix over the $\ell^{\rm th}$ roots of unity for any $\ell$ dividing $k$, provided that any prime $p$ dividing $k$ also divides $\ell$. We prove that a $\mathbb{Z}_{p^s}$-additive code with $p$ a prime number is isomorphic as a group to a BH-code over $\mathbb{Z}_{p^s}$ and the image of this BH-code under the Gray map is a BH-code over $\mathbb{Z}_p$ (binary Hadamard code for $p=2$). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.
