Quasi optimal anticodes: structure and invariants

广义权：一种反编码方法

Generalized weights: an anticode approach

在本文中，我们研究广义重量作为一个代数不变的代码。我们首先描述的汉明和秩度量的反码，证明特别是最佳的反码的秩度量符合Frobenius闭空间。然后，我们描述了广义汉明和秩权重的代码的交叉点的代码与最佳的反码在各自的度量。受这种描述的启发，我们提出了一个新的代数不变式，我们称之为“Delsarte广义权重”，Delsarte秩度量码的基础上的最佳反码的矩阵。我们证明了我们的不变量精化了Kurihara，Matsumoto和Uyematsu提出的Gabidulin码的广义秩权，并建立了Delsarte广义权的一系列性质.特别是，我们刻画Delsarte最优码和反码的广义权重。我们还提出了一个新的代数不变量的对偶理论，证明了一个码的Delsarte广义重量完全确定的Delsarte广义重量的对偶码。我们的结果推广了Gabidulin码的广义秩权理论。最后，我们证明了类似的Gabidulin码的一个定理魏，证明了他们的广义秩重量的特点的Gabidulin秩度量码的最坏情况下的安全下降。

弱组合格栅与环线性反码

Weak Composition Lattices and Ring-Linear Anticodes

本文研究了环上Lee度量反码的结构，并建立了其与弱组合格的双射关系，进而引入新的编码理论不变量。

Tensor Codes and their Invariants

In 1991, Roth introduced a natural generalization of rank metric codes, namely tensor codes. The latter are defined to be subspaces of $r$-tensors where the ambient space is endowed with the tensor rank as a distance function. In this work, we describe the general class of tensor codes and we study their invariants that correspond to different families of anticodes. In our context, an anticode is a perfect space that has some additional properties. A perfect space is one that is spanned by tensors of rank 1. Our use of the anticode concept is motivated by an interest in capturing structural pr…

Rank-Metric Codes and $q$-Polymatroids

This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.

和秩度量码

Sum-rank metric codes

本章系统阐述和秩度量码的数学理论，涵盖不变量、最优反码、MSRD 码及 F_{q^m} 线性码等主要成果。