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Parity codes

Published online by Cambridge University Press:  15 March 2005

Paulo E. D. Pinto ,
Fábio Protti  and
Jayme L. Szwarcfiter
Paulo E. D. Pinto
Affiliation:
Universidade Estadual do Rio de Janeiro, Instituto de Matemática e Estatística, RJ, Brasil; pauloedp@ime.uerj.br
Fábio Protti
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática and NCE, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brasil; fabiop@nce.ufrj.br
Jayme L. Szwarcfiter
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brasil; jayme@nce.ufrj.br
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Abstract

Motivated by a problem posed by Hamming in 1980, we define even codes. They are Huffman type prefix codes with the additional property of being able to detect the occurrence of an odd number of 1-bit errors in the message. We characterize optimal even codes and describe a simple method for constructing the optimal codes. Further, we compare optimal even codes with Huffman codes for equal frequencies. We show that the maximum encoding in an optimal even code is at most two bits larger than the maximum encoding in a Huffman tree. Moreover, it is always possible to choose an optimal even code such that this difference drops to 1 bit. We compare average sizes and show that the average size of an encoding in a optimal even tree is at least 1/3 and at most 1/2 of a bit larger than that of a Huffman tree. These values represent the overhead in the encoding sizes for having the ability to detect an odd number of errors in the message. Finally, we discuss the case of arbitrary frequencies and describe some results for this situation.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 263 - 278
Copyright
© EDP Sciences, 2005

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