论文标题
在特征零中支持GABIDULIN代码的受约束发电机矩阵
Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero
论文作者
论文摘要
每当基础场扩展的Galois组都是循环时，Augot等人最近就构建了特征零字段上的Gabidulin代码。同时，由于在分布式计算中的应用，对芦苇 - 固体和加贝杜蛋白代码的稀疏发生器矩阵的兴趣最近增加了。特别是，与有限场上的Gabidulin代码的最稀疏的可能发生器矩阵存在相关的零条目相交的一定条件是必要且足够的。在本文中，我们通过证明特征零字段上的Gabidulin代码也需要相同的条件来完成图片。我们的证明基于有限的现场案例并扩展了工具，并将它们与自动形态上的Schwartz-Zippel引理的变体结合在一起，并提供了一种简单的随机构造算法，其成功的可能性可以任意接近一个。此外，讨论了低级矩阵恢复的潜在应用。
Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed-Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero. Our proof builds upon and extends tools from the finite field case, combines them with a variant of the Schwartz-Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed.
