A Novel Representation for Permutations

In this paper, we present a variable-length binary code for permutations of degree n , where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at t...

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Vydáno v:IEEE transactions on information theory Ročník 67; číslo 3; s. 1920
Hlavní autoři: Won-Ho Ri, Ok-Hyon Song
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 01.03.2021
Témata:
Binary codes
Codes
Decoding
Enumeration
Lookup tables
Permutations
ISSN:0018-9448, 1557-9654
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Shrnutí:In this paper, we present a variable-length binary code for permutations of degree n , where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover’s enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary [Formula Omitted] in a straightforward way.
Bibliografie:ObjectType-Article-1
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content type line 14
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2020.3048905