Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently. In this work, we are interested in the maximum size of 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 by analyzing the independence number of the auxiliary graph. The idea is widely used in various cases and our contribution in this section is enumerating the number of triangles in the auxiliary graph and showing that it is small enough. Next, we design permutation codes correcting multiple deletions with a decoding algorithm. 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. Our construction is based on a new mapping which yields a new connection between permutation codes in the Hamming metric and permutation codes in various metrics. Furthermore, we construct permutation codes that correct multiple bursts of deletions using this new mapping. Finally, we extend the new mapping for multi-permutations and construct the best-known multi-permutation codes in Ulam metric.

翻译：近年来，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。我们首先通过分析辅助图的独立数，改进了这类置换码最大规模的Gilbert--Varshamov界。这一思路在多种场景中被广泛使用，我们在本节的主要贡献是计算了辅助图中三角形的数量，并证明该数量足够小。接着，我们设计了能够纠正多重删除且具备解码算法的置换码。具体而言，所构造的置换码能够纠正$t$次删除，其冗余度至多为$(3t-1) \log n+o(\log n)$比特，其中$n$为码长。我们的构造基于一种新的映射方法，该映射揭示了Hamming度量下的置换码与多种度量下的置换码之间的新联系。此外，我们利用这一新映射构造了能够纠正多重突发删除的置换码。最后，我们将该新映射推广至多重置换，构造了Ulam度量下目前最优的多重置换码。