论文标题
在$ z_ {p^r} z_ {p^r} z_ {p^s} $ - 加法循环代码
On $Z_{p^r}Z_{p^r}Z_{p^s}$-Additive Cyclic Codes
论文作者
论文摘要
在本文中，我们介绍了$ \ mathbb {z} _ {p^r} \ mathbb {z} _ {p^r} \ mathbb {z} _ {p^s} $ - 添加性循环代码$ r \ leq s $。这些代码可以被识别为$ \ mathbb {z} _ {p^s} [x] $ - $ \ MATHBB {z} _ {z} _ {p^r} [x]/\ langle x^α-1 \ langle x^α-1\ rangle \ rangle \ rangle \ times \ times \ times \ times \ times \ mathb {z} _ {z {p^lang X^β-1 \ rangle \ times \ times \ mathbb {z} _ {p^s} [x]/\ langle x^γ-1 \ rangle $。我们确定了该代码系列的发电机多项式和最小生成集。案例已经完成了一些以前的工作，$ p = 2 $，$ r = s = 1 $，$ r = s = 2 $，$ r = 1，s = 2 $。但是，我们表明，在这些先前的作品中，这些代码的分类不完整，本文中的陈述完成了此类分类。我们还讨论了可分离$ \ Mathbb {z} _ {p^r} \ Mathbb {z} _ {p^r} \ Mathbb {z} _ {p^s} $ - 添加性环状代码并确定其发电机polynomials的结构。此外，我们还研究了$ \ Mathbb {z} _ {p^s} [x] $ - subsodules的双重性。作为应用程序，我们提供一些示例并构建一些最佳二进制代码。
In this paper, we introduce $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes for $r\leq s$. These codes can be identified as $\mathbb{Z}_{p^s}[x]$-submodules of $\mathbb{Z}_{p^r}[x]/\langle x^α-1\rangle \times \mathbb{Z}_{p^r}[x]/\langle x^β-1\rangle\times \mathbb{Z}_{p^s}[x]/\langle x^γ-1\rangle$. We determine the generator polynomials and minimal generating sets for this family of codes. Some previous works has been done for the case $p=2$ with $r=s=1$, $r=s=2$, and $r=1,s=2$. However, we show that in these previous works the classification of these codes were incomplete and the statements in this paper complete such classification. We also discuss the structure of separable $\mathbb{Z}_{p^r}\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes and determine their generator polynomials. Further, we also study the duality of $\mathbb{Z}_{p^s}[x]$-submodules. As applications, we present some examples and construct some optimal binary codes.
