论文标题
z_pz_ {p^2}的构建和线性性 - 线性广义HADAMARD代码
Construction and Linearity of Z_pZ_{p^2}-Linear Generalized Hadamard Codes
论文作者
论文摘要
$ \ z_p \ z_ {p^2} $ - 添加代码是$ \ z_p^{α_1} \ times \ z__ {p^2}^{α_2} $的子组，并且可以看作是$ \ z_p $ a $α__2= 0 $ al的$α__2= 0 $， $α_1= 0 $，或$ \ z_2 \ z_4 $ - addive代码时$ p = 2 $。 a $ \ z_p \ z_ {p^2} $ - 线性广义hadamard（gh）代码是$ \ z_p $上的GH代码，它是$ \ z_p \ z__ {p^2} $ - 附加代码的灰色地图图像。在本文中，我们将一些已知结果推广为$ \ z_p \ z_ {p^2} $ - $ p = 2 $的线性gh代码在任何$ p \ geq 3 $ prime时$α_1\ neq 0 $。首先，我们给出了$ \ z_p \ z_ {p^2} $的递归结构 - $ t_1，t_2 \ geq 1 $。然后，我们显示哪种类型相应的$ \ z_p \ z_ {p^2} $ - 线性GH代码在$ \ z_p $上是非线性的。最后，根据一些计算结果，我们看到，与$ \ z_4 $ -linear GH代码不同，当$ p \ geq 3 $ prime时，$ \ z_ {p^2} $ - 线性GH代码不包括在$ \ z_p \ z____________________ {p^2} $ -linearear gh codes codes codes cop $ al ynecy $ a $ a $α________。
The $\Z_p\Z_{p^2}$-additive codes are subgroups of $\Z_p^{α_1} \times \Z_{p^2}^{α_2}$, and can be seen as linear codes over $\Z_p$ when $α_2=0$, $\Z_{p^2}$-additive codes when $α_1=0$, or $\Z_2\Z_4$-additive codes when $p=2$. A $\Z_p\Z_{p^2}$-linear generalized Hadamard (GH) code is a GH code over $\Z_p$ which is the Gray map image of a $\Z_p\Z_{p^2}$-additive code. In this paper, we generalize some known results for $\Z_p\Z_{p^2}$-linear GH codes with $p=2$ to any $p\geq 3$ prime when $α_1 \neq 0$. First, we give a recursive construction of $\Z_p\Z_{p^2}$-additive GH codes of type $(α_1,α_2;t_1,t_2)$ with $t_1,t_2\geq 1$. Then, we show for which types the corresponding $\Z_p\Z_{p^2}$-linear GH codes are non-linear over $\Z_p$. Finally, according to some computational results, we see that, unlike $\Z_4$-linear GH codes, when $p\geq 3$ prime, the $\Z_{p^2}$-linear GH codes are not included in the family of $\Z_p\Z_{p^2}$-linear GH codes with $α_1\not =0$.
