Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently owing to their applications. In this work, we are interested in the maximum size of the permutation codes in the Ulam metric and aim to design permutation codes that can correct multiple deletions with efficient decoding algorithms. We first present an improvement on the Gilbert--Varshamov bound of the maximum size of these permutation codes which is the best-known lower bound. Next, we focus on designing permutation codes in the Ulam metric with a decoding algorithm. These constructed codes are the best-known permutation codes that can correct multiple deletions. In particular, the constructed permutation codes can correct $t$ deletions with at most $(3t-1) \log n+o(\log n)$ bits of redundancy where $n$ is the length of the code. Finally, we provide an efficient decoding algorithm for our constructed permutation codes.

翻译：近年来，由于其在应用中的重要性，能够纠正多重删除的Ulam度量下的置换码得到了广泛研究。本文关注Ulam度量下置换码的最大规模，并致力于设计能够纠正多重删除且具有高效解码算法的置换码。我们首先改进了这类置换码最大规模的Gilbert--Varshamov界，这是目前已知的最佳下界。接着，我们重点设计具有解码算法的Ulam度量置换码。所构造的码是目前已知的能够纠正多重删除的最佳置换码。具体而言，所构造的置换码能够纠正$t$个删除，其冗余度至多为$(3t-1) \log n+o(\log n)$比特，其中$n$为码长。最后，我们为所构造的置换码提供了一种高效解码算法。