论文标题
美元
$\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-Additive Generalized Hadamard Codes
论文作者
论文摘要
$ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {p^s} $ - 添加代码是$ \ mathbb {z} \mathbb{Z}_{p^2}^{α_2} \times \cdots \times \mathbb{Z}_{p^s}^{α_s}$, and can be seen as linear codes over $\mathbb{Z}_p$ when $α_i=0$ for all $i \in \ {2，\ dots，s \} $，a $ \ mathbb {z} _ {p^s} $ - $α_i= 0 $ for ALL $ i \ in \ in \ in \ in \ in \ {1，\ dots，s-1 \}，s-1 \} $，或a $ \ \ mathbb {z p \ n 2 $ s = 2 $或$ \ mathbb {z} _2 \ mathbb {z} _4 $ addive代码时代码时代码。 a $ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {z} _ {p^s} $ - 线性概括hadamard（gh）代码是$ \ mathbb {z} _p $ a的GH代码，是一个是gh {z} _p $，是灰色映射的图像。 $ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {p^s} $ - 添加代码。在本文中，我们将一些已知结果概括为$ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {p^s} $ - 线性-Linarear GH代码，带有$ p $ prime和$ s \ s \ geq 2 $。首先，我们给出$ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {p^s} $ - 添加性GH代码$（α_1，\ dots; t_2，\ dots，t_ {s-1} \ geq 0 $和$ t_s \ geq1 $。然后，我们展示哪种类型的$ \ mathbb {z} _p \ mathbb {z} _ {p^2} \ dots \ mathbb {z} _ {p^s} $ - 线性gh代码超过$ \ mthbb {z} _ _ _ _p $。我们还要在非线性时计算内核及其尺寸。
The $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_p^{α_1} \times \mathbb{Z}_{p^2}^{α_2} \times \cdots \times \mathbb{Z}_{p^s}^{α_s}$, and can be seen as linear codes over $\mathbb{Z}_p$ when $α_i=0$ for all $i \in \{2,\dots, s\}$, a $\mathbb{Z}_{p^s}$-additive code when $α_i=0$ for all $i \in \{1,\dots, s-1\}$ , or a $\mathbb{Z}_p\mathbb{Z}_{p^2}$-additive code when $s=2$, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $p=2$ and $s=2$. A $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-additive code. In this paper, we generalize some known results for $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH codes with $p$ prime and $s\geq 2$. First, we give a recursive construction of $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots \mathbb{Z}_{p^s}$-additive GH codes of type $(α_1,\dots,α_s;t_1,\dots,t_s)$ with $t_1\geq 1, t_2,\dots,t_{s-1}\geq 0$, and $t_s\geq1$. Then, we show for which types the corresponding $\mathbb{Z}_p\mathbb{Z}_{p^2}\dots\mathbb{Z}_{p^s}$-linear GH codes are nonlinear over $\mathbb{Z}_p$. We also compute the kernel and its dimension whenever they are nonlinear.
