Parity codes

Paulo E. D. Pinto; Fábio Protti; Jayme L. Szwarcfiter

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2005)

  • Volume: 39, Issue: 1, page 263-278
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Pinto, Paulo E. D., Protti, Fábio, and Szwarcfiter, Jayme L.. "Parity codes." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 39.1 (2005): 263-278. <http://eudml.org/doc/244903>.

@article{Pinto2005,
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.},
author = {Pinto, Paulo E. D., Protti, Fábio, Szwarcfiter, Jayme L.},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {Hamming-Huffman code; even code; prefix code; optimal even code},
language = {eng},
number = {1},
pages = {263-278},
publisher = {EDP-Sciences},
title = {Parity codes},
url = {http://eudml.org/doc/244903},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Pinto, Paulo E. D.
AU - Protti, Fábio
AU - Szwarcfiter, Jayme L.
TI - Parity codes
JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 1
SP - 263
EP - 278
AB - 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.
LA - eng
KW - Hamming-Huffman code; even code; prefix code; optimal even code
UR - http://eudml.org/doc/244903
ER -

References

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  11. [11] R.L. Milidiu and E.S. Laber, Improved Bounds on the Inefficiency of Length Restricted Codes. Algorithmica 31 (2001) 513–529. Zbl1012.94008
  12. [12] A. Turpin and A. Moffat, Practical length-limited coding for large alphabets. Comput. J. 38 (1995) 339–347. 
  13. [13] P.E.D Pinto, F. Protti and J.L. Szwarcfiter, A Huffman-Based Error Detection Code, in Proc. of the Third International Workshop on Experimental and Efficient Algorithms (WEA 2004), Angra dos Reis, Brazil, 2004. Lect. Notes Comput. Sci. 3059 (2004) 446–457. 
  14. [14] E.S. Schwartz, An Optimum Encoding with Minimal Longest Code and Total Number of Digits. Inform. Control 7 (1964) 37–44. Zbl0116.35303

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