The covering radius is a fundamental property of linear codes that characterizes the trade-off between storage and access in linear data-query protocols. The generalized covering radius was recently defined by Elimelech and Schwartz for applications in joint-recovery of linear data-queries. In this work we extend a known bound on the ordinary covering radius to the generalized one for all codes satisfying the chain condition- a known condition which is satisfied by most known families of codes. Given a generator matrix of a special form, we also provide an algorithm which finds codewords which cover the input vectors within the distance specified by the bound. For the case of Reed-Muller codes we provide efficient construction of such generator matrices, therefore providing a faster alternative to a previous generalized covering algorithm for Reed-Muller codes.

翻译：覆盖半径是线性码的基本性质，它刻画了线性数据查询协议中存储与访问之间的权衡关系。Elimelech和Schwartz近期针对线性数据查询的联合恢复应用定义了广义覆盖半径。本文针对所有满足链式条件（已知大多数常见码族均满足该条件）的码，将普通覆盖半径的已知界推广至广义覆盖半径。给定特定形式的生成矩阵，我们提出了一种算法，能够在界所规定的距离内找到覆盖输入向量的码字。对于Reed-Muller码，我们给出了此类生成矩阵的高效构造方法，从而为此前针对Reed-Muller码的广义覆盖算法提供了更快速的替代方案。