论文标题
线性化的REED - 固体代码，具有支撑受限的生成器矩阵和多源网络编码中的应用
Linearized Reed-Solomon Codes with Support-Constrained Generator Matrix and Applications in Multi-Source Network Coding
论文作者
论文摘要
线性化的Reed-Solomon（LRS）代码是基于偏斜多项式的评估代码。他们在总和度量中实现了界限的单例，因此被称为最大总和距离（MSRD）代码。在这项工作中，我们提供了具有支持约束的发电机矩阵的MSRD代码的必要条件。支持约束的条件与MDS代码和MRD代码的条件相同。 $ [n，k] _ {q^m} $ lrs代码带有支撑受限的生成器矩阵的所需字段大小为$ q \ geq \ ell+1 $和$ m \ geq \ geq \ geq \ max_ {l \ in [\ ell]} \ in [\ ell]} \ in [\ ell]} \ {k-1+\ log__qk，n_qk，n_qk，n_l \ el $ $ l $ -th块结果的特殊情况与REED - 固体代码和Gabidulin代码的已知结果相吻合。对于不满足必要条件的支持约束，我们得出了其发电机矩阵满足约束的代码的最大总和量距离。可以从具有足够大的字段大小的LRS代码的子代码构建此类代码。此外，作为网络编码中的应用程序，可以将条件用作整数编程问题中的约束，以设计分布式多源网络的分布式LRS代码。
Linearized Reed-Solomon (LRS) codes are evaluation codes based on skew polynomials. They achieve the Singleton bound in the sum-rank metric and therefore are known as maximum sum-rank distance (MSRD) codes. In this work, we give necessary and sufficient conditions for the existence of MSRD codes with a support-constrained generator matrix. The conditions on the support constraints are identical to those for MDS codes and MRD codes. The required field size for an $[n,k]_{q^m}$ LRS codes with support-constrained generator matrix is $q\geq \ell+1$ and $m\geq \max_{l\in[\ell]}\{k-1+\log_qk, n_l\}$, where $\ell$ is the number of blocks and $n_l$ is the size of the $l$-th block. The special cases of the result coincide with the known results for Reed-Solomon codes and Gabidulin codes. For the support constraints that do not satisfy the necessary conditions, we derive the maximum sum-rank distance of a code whose generator matrix fulfills the constraints. Such a code can be constructed from a subcode of an LRS code with a sufficiently large field size. Moreover, as an application in network coding, the conditions can be used as constraints in an integer programming problem to design distributed LRS codes for a distributed multi-source network.
