论文标题
关于等效线性代码的船体变化问题
On Hull-Variation Problem of Equivalent Linear Codes
论文作者
论文摘要
交叉点$ {\ bf c} \ bigCap {\ bf c}^{\ perp} $（$ {\ bf c} \ bigcap {\ bf c}^{\ bf c}^{\ perp_h} $ c}^{\ perp} $（Hermitian dual dual $ {\ bf c}^{\ perp_h} $）称为此代码的欧几里得人（Hermitian）船体。当线性代码$ {\ bf c} $转换为等效代码$ {\ bf v} \ cdot {\ bf c} $时，考虑赫尔差异问题是很自然的。在本文中，我们将最大船体维度引入了相对于等效转换的线性代码的不变。然后研究了最大船体尺寸的一些基本特性。我们证明，对于满足$ 0 \ leq H \ leq n-1 $的非负整数$ h $，线性$ [2n，n] _q $ self-Dual代码等于线性$ h $ dimension hull hull代码。在相反的方向上，我们证明了$ {\ bf f} _ {2^s} $满足$ d \ geq 2 $和$ d^{\ perp} \ geq 2 $的线性LCD代码等同于在弱条件下线性的一维Hull代码。还构建了$ {\ bf f} _3 $的LCD NegacyClic代码和LCD BCH代码的几个新系列。我们的方法可以应用于广义的芦苇 - 固体代码和广义的扭曲的芦苇 - 固体代码，以构建任意尺寸船体MDS代码。构建了一些新的纠缠辅助量子误差（EAQEC）代码，包括MDS和MDS EAQEC代码。小型领域上的许多EAQEC代码都是由最佳的Hermitian自动偶联代码构建的。
The intersection ${\bf C}\bigcap {\bf C}^{\perp}$ (${\bf C}\bigcap {\bf C}^{\perp_h}$) of a linear code ${\bf C}$ and its Euclidean dual ${\bf C}^{\perp}$ (Hermitian dual ${\bf C}^{\perp_h}$) is called the Euclidean (Hermitian) hull of this code. It is natural to consider the hull-variation problem when a linear code ${\bf C}$ is transformed to an equivalent code ${\bf v} \cdot {\bf C}$. In this paper we introduce the maximal hull dimension as an invariant of a linear code with respect to the equivalent transformations. Then some basic properties of the maximal hull dimension are studied. We prove that for a nonnegative integer $h$ satisfying $0 \leq h \leq n-1$, a linear $[2n, n]_q$ self-dual code is equivalent to a linear $h$-dimension hull code. On the opposite direction we prove that a linear LCD code over ${\bf F}_{2^s}$ satisfying $d\geq 2$ and $d^{\perp} \geq 2$ is equivalent to a linear one-dimension hull code under a weak condition. Several new families of LCD negacyclic codes and LCD BCH codes over ${\bf F}_3$ are also constructed. Our method can be applied to the generalized Reed-Solomon codes and the generalized twisted Reed-Solomon codes to construct arbitrary dimension hull MDS codes. Some new entanglement-assisted quantum error-correction (EAQEC) codes including MDS and almost MDS EAQEC codes are constructed. Many EAQEC codes over small fields are constructed from optimal Hermitian self-dual codes.
