arXiv 2102.069952021-02-13cs.ITcs.DMmath.IT

Galois hulls of cyclic serial codes over a finite chain ring

Sarra Talbi, Aicha Batoul, Alexandre Fotue Tabue, Edgar Martínez-Moro

备注：: Submitted to Finite Fields and Their Applications

英文摘要：

In this paper, we explore some properties of Galois hulls of cyclic serial codes over a chain ring and we devise an algorithm for computing all the possible parameters of the Euclidean hulls of that codes. We also establish the average $p^r$-dimension of the Euclidean hull, where $\mathbb{F}_{p^r}$ is the residue field of $R$, and we provide some results of its relative growth.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

On MDS Codes With Galois Hulls of Arbitrary Dimensions

The Galois hulls of linear codes are a generalization of the Euclidean and Hermitian hulls of linear codes. In this paper, we study the Galois hulls of (extended) GRS codes and present several new constructions of MDS codes with Galois hulls of arbitrary dimensions via (extended) GRS codes. Two general methods of constructing MDS codes with Galois hulls of arbitrary dimensions by Hermitian or general Galois self-orthogonal (extended) GRS codes are given. Using these methods, some MDS codes with larger dimensions and Galois hulls of arbitrary dimensions can be obtained and relatively strict con…

2022-09-19cs.ITmath.IT

New Galois Hulls of GRS Codes and Application to EAQECCs

Galois hulls of linear codes have important applications in quantum coding theory. In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions. We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes. Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results.

2021-07-31cs.ITmath.IT

有限域上线性码的Galois壳

Galois Hulls of Linear Codes over Finite Fields

有限域$\mathbb{F}_q$上的$[n,k]$线性码$C$的$\ell$ -Galois壳$h_{\ell}(C)$是$C$和$C^{{\bot}_{\ell}}$的交集，其中$C^{\bot_{\ell}}$表示Fan和Zhang（2017）引入的$C$的$\ell$ -Galois对偶。$\ell$ - Galois LCD代码是一个线性代码$C$，带有$h_{\ell}(C) = 0$。本文证明了线性码的$\ell$ -Galois包的维数在置换等价下是不变的，并给出了一种由码的生成矩阵计算$\ell$ -Galois包维数的方法.此外，我们还得到了三元码的$\ell$ -Galois壳的维数在单项式等价下也是不变的。如果$q>4$，则码的$l$ -伽罗瓦外壳的维数在单项式等价下不是不变的。证明了在$\mathbb F_{q}$上的每个$[n,k]$线性码都等价于对任何$q>4$的一个$\ell$ -Galois LCD码.我们的结论是，如果对任何$q>4$都存在$\mathbb F_{q}$上的$[n,k]$线性码，那么对任何$0\le \ell\le e-1$都存在具有相同参数的$\ell$ LCD码，其中$q=p^e$对某个素数$p$ .作为应用，我们刻画了有限域上矩阵乘积码的$\ell$ -Galois壳.

2018-09-21cs.ITmath.IT

On Galois hulls of linear codes and new entanglement-assisted quantum error-correcting codes

The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls of linear codes is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes, and Galois linear complementary dual codes are characterized. Then, we propose explicit methods to construct Galois self-orthogonal codes of larger length from given Galois self-orthogonal co…

2023-01-11cs.ITmath.IT

MDS码的Galois壳及其量子纠错

Galois hulls of MDS codes and their quantum error correction

线性码的包在量子信息和编码理论中起着重要的作用。本文通过研究有限域Fq上广义Reed-Solomon（GRS）码和扩展GRS码的Galois壳，给出了几类新的具有任意维数Galois壳的MDS码，这类MDS码是以前没有得到的.他们中的一些人概括了文献[13]中的那些。因此，使用这些MDS码与伽罗瓦壳的任意尺寸，我们构建了9个新的家庭MDS纠缠辅助量子纠错码（EAQECCs）与灵活的参数。

2020-04-05cs.ITmath.ITquant-ph

广义Roth-Lempel码的Galois壳及其在EAQECCs中的应用

Galois Hulls of Generalized Roth-Lempel Codes and Their Applications to EAQECCs

School of Mathematics, Nanjing Normal University（南京师范大学数学学院）

本文提出基于归一化拉格朗日系数的通用方法，构造具有指定Galois壳维数的广义Roth-Lempel码，并应用于纠缠辅助量子纠错码，获得MDS/AMDS码及最大纠缠EAQECCs。

2026-09-18cs.ITmath.IT

更多相关论文收起更多论文

Galois Hull Dimensions of Gabidulin Codes

For a prime power $q$, an integer $m$ and $0\leq e\leq m-1$ we study the $e$-Galois hull dimension of Gabidulin codes $G_k(\boldsymbolα)$ of length $m$ and dimension $k$ over $\mathbb{F}_{q^m}$. Using a self-dual basis $\boldsymbolα$ of $\mathbb{F}_{q^m}$ over $\mathbb{F}_q$, we first explicitly compute the hull dimension of $G_k(\boldsymbolα)$. Then a necessary and sufficient condition of $G_k(\boldsymbolα)$ to be linear complementary dual (LCD), self-orthogonal and self-dual will be provided. We prove the existence of $e$-Galois (where $e=\frac{m}{2}$) self-dual Gabidulin codes of length $m$…

2022-11-09cs.ITmath.IT

仿射代数环上常循环码的Galois包络

Galois hulls of constacyclic codes over affine algebra rings

该研究针对仿射代数环上的λ-常循环码，定义了k-Galois内积，推导了其k-Galois对偶码、包络的生成元形式与包络维数公式，给出k-Galois LCD码的判定条件，并展示了其在纠缠辅助量子纠错码构造中的应用。

2024-12-11cs.ITmath.IT

Intersections of linear codes and related MDS codes with new Galois hulls

Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. In this paper, we investigate general properties of linear codes associated with $σ$ duals for $σ\in\mathrm{SLAut}(\mathbb{F}_{q}^{n})$. We show that the dimension of the intersection of two linear codes can be determined by generator matrices of such codes and their $σ$ duals. We also show that the dimension of $σ$ hull of a linear code can be determined by a generator matrix of it or its $σ$ dual. We give a characterization on $σ$ dual and $σ$ hull…

2023-03-24cs.ITmath.IT

多扭转码Galois核的生成多项式矩阵

Generator polynomial matrices of the Galois hulls of multi-twisted codes

本文研究了有限域上多扭转码的Galois核，通过将生成多项式矩阵与准循环码关联，推导出核的维数公式和GPM公式，并建立了自正交与线性互补对偶码的等价条件。

2023-05-24cs.ITmath.IT

奇特征下自正交码与完全自正交码的采样算法

Algorithms for Sampling Self-Orthogonal and Totally Self-Orthogonal Codes in Odd Characteristic

King’s College London（伦敦国王学院）；SandboxAQ（SandboxAQ公司）；Royal Holloway, University of London（伦敦大学皇家霍洛威学院）；Aalto University（阿尔托大学）

本文提出在奇特征有限域上均匀采样任意壳维数线性码的算法，首次实现任意码率自正交码采样，并研究完全正交性以支持PCE密码方案。

2026-10-07cs.ITmath.IT

具有大维度的 Galois 自正交 MDS 码

Galois self-orthogonal MDS codes with large dimensions

本文给出 GRS/EGRS 码及其扩展码 Galois 自正交的充要条件，分类构造维度更大的新 Galois 自正交 MDS 码，并导出任意维度 Galois hull 的 MDS 码及量子码。

2024-12-06cs.ITmath.IT

Several classes of Galois self-orthogonal MDS codes and related applications

Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. $e$-Galois self-orthogonal codes are generalizations of Euclidean self-orthogonal codes ($e=0$) and Hermitian self-orthogonal codes ($e=\frac{h}{2}$ and $h$ is even). In this paper, we propose two general methods to construct $e$-Galois self-orthogonal (extended) generalized Reed-Solomon (GRS) codes. As a consequence, eight new classes of $e$-Galois self-orthogonal (extended) GRS codes with odd $q$ and $2e\mid h$ are obtained. Based on the Galois dual of a code, we also study its punctured and shortened codes. As applic…

2023-02-09cs.ITmath.IT

相对壳与量子码

Relative hulls and quantum codes

本文研究 $q$ 元码的相对壳维数，证明 $q>2$ 时可通过等价码逐步减小维数至下界，并给出增大条件，进而构造新的纠缠辅助量子最大距离可分码。

2023-12-23cs.ITmath.ITmath.RA