Efficient encoding and decoding algorithm for a class of perfect single-deletion-correcting permutation codes

TL;DR

This paper presents a new, direct proof of Levenshtein's perfect single-deletion-correcting permutation codes and introduces efficient algorithms for encoding and decoding within this class.

Contribution

It offers a novel proof independent of Varshamov-Tenengolts codes and develops more efficient algorithms for permutation code correction.

Findings

New direct proof of Levenshtein's code construction

Efficient encoding algorithm for single-deletion correction

Efficient decoding algorithm for single-deletion correction

Abstract

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.