Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Minhan Gao, Kenneth W. Shum

TL;DR

This paper gives an independent and more direct proof of Levenshtein's result that does not depend on the Varshamov-Tenengolts code, and devise efficient encoding and decoding algorithms that correct one deletion.

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.

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

TL;DR

This paper gives an independent and more direct proof of Levenshtein's result that does not depend on the Varshamov-Tenengolts code, and devise efficient encoding and decoding algorithms that correct one deletion.

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.

Paper Structure

This paper contains 9 sections, 8 theorems, 91 equations, 4 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Permutation codes with deletion metric
  3. Transposition and block transposition
  4. A representation of permutations using suffix block transpositions
  5. An alternate description of Levenshtein's permutation codes
  6. Perfect 1-deletion permutation code
  7. Proof of Theorem \ref{['thm:distint_values']}
  8. Encoding and decoding algorithm
  9. Concluding Remarks

Key Result

Theorem 1

Let $V_n$ denote the set of vectors as in eq:V_n. The map in eq:inverse_rep is a bijection from the set $V_n$ defined in eq:V_n to the symmetric group $S_n$.

Figures (4)

  • Figure 1: Reverse calculations codeword is $\rho^{-1} = (3,0,5,1,2,4)$, whose inverse is $\rho = (1,3,4,0,5,2)$.
  • Figure 2: The effect of a symbol deletion.
  • Figure 3: Illustration of Equation \ref{['eq:transposition_product']}
  • Figure 4: Periodic presentation of the permutation $\rho = (0,4,1,3,2)$. The locations with the largest value is highlighted.

Theorems & Definitions (20)

  • Example 1
  • Example 2
  • Theorem 1
  • Definition 1
  • Example 3
  • Example 4
  • Theorem 2
  • proof
  • Remark 1
  • Example 5
  • ...and 10 more