A permutation code is a nonlinear code whose codewords are permutation of a set of symbols. We consider the use of permutation code in the deletion channel, and consider the symbol-invariant error model, meaning that the values of the symbols that are not removed are not affected by the deletion. In 1992, Levenshtein gave a construction of perfect single-deletion-correcting permutation codes that attain the maximum code size. Furthermore, he showed in the same paper that the set of all permutations of a given length can be partitioned into permutation codes so constructed. This construction relies on the binary Varshamov-Tenengolts codes. In this paper we give an independent and more direct proof of Levenshtein's result that does not depend on the Varshamov-Tenengolts code. Using the new approach, we devise efficient encoding and decoding algorithms that correct one deletion.

翻译：置换码是一种非线性码，其码字为符号集合的排列。我们研究置换码在删除信道中的应用，并考虑符号不变误差模型，即未被删除的符号值不受删除操作影响。1992年，Levenshtein提出了一种达到最大码集规模的完美单删除校正置换码构造方法。此外，他在同一篇论文中证明了给定长度的所有置换集合可被划分为此类构造的置换码。该构造依赖于二进制Varshamov-Tenengolts码。本文提出了一种独立且更直接的证明方法，该证明不依赖于Varshamov-Tenengolts码。基于新方法，我们设计了能够校正单删除的高效编码与解码算法。