论文标题
$ \ mathbb {f} _p $ addive genter genter gentrized hadamard代码的等级和内核
Rank and Kernel of $\mathbb{F}_p$-Additive Generalised Hadamard Codes
论文作者
论文摘要
矢量空间$ \ mathbb {f} _q^n $的子集如果是$ k $ addive的，则如果它是子字段$ k \ subseteq \ subseteq \ mathbb {f} _q $上的线性空间。令$ q = p^e $，$ p $ prime和$ e> 1 $。建立了$ \ mathbb {f} _p $ - addive的广义hadamard（gh）代码内核的等级和维度的界限。对于这些边界内内核的特定等级和尺寸，构建了$ \ mathbb {f} _p $ - addive GH代码。此外，对于$ e = 2 $的情况，这表明给定的边界很紧，并且可以为所有允许等级和这些边界之间的内核的所有允许等级和尺寸构造$ \ mathbb {f} _p $ addive gh code。最后，我们还证明这些代码相对于痕迹赫尔米亚内部产品是自动的，并生成纯量子代码。
A subset of a vector space $\mathbb{F}_q^n$ is $K$-additive if it is a linear space over the subfield $K\subseteq \mathbb{F}_q$. Let $q=p^e$, $p$ prime, and $e>1$. Bounds on the rank and dimension of the kernel of generalised Hadamard (GH) codes which are $\mathbb{F}_p$-additive are established. For specific ranks and dimensions of the kernel within these bounds, $\mathbb{F}_p$-additive GH codes are constructed. Moreover, for the case $e=2$, it is shown that the given bounds are tight and it is possible to construct an $\mathbb{F}_p$-additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.
