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In this paper, we discussed binary fuzzy codes over a vector space F2n by relating classical codes with the probability of a binary symmetric channel (BSC) for receiving a sent codeword correctly. We used the weight of error patterns between a received word and the possible sent codewords to define fuzzy words over n-dimensional vector space F2n, and used it to define binary fuzzy codes. We also added some properties of Hamming distance of binary fuzzy codes, and the bounds of a Hamming distance of binary fuzzy codes for p=1/r, where r3, and rZ+, are determined. Finding Hamming distance of binary fuzzy codes is used for decoding sent messages on a BSC.


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Binary Fuzzy Codes and Some Properties of Hamming Distance of Fuzzy Codes

Show Author's information Mezgebu Manmekto Gereme1,2( )Jejaw Demamu1,2Berhanu Assaye Alaba1
Department of Mathematics, Bahir Dar University, Bahir Dar 6000, Ethiopia
Department of Mathematics, Debark University, Debark 6200, Ethiopia

Abstract

In this paper, we discussed binary fuzzy codes over a vector space F2n by relating classical codes with the probability of a binary symmetric channel (BSC) for receiving a sent codeword correctly. We used the weight of error patterns between a received word and the possible sent codewords to define fuzzy words over n-dimensional vector space F2n, and used it to define binary fuzzy codes. We also added some properties of Hamming distance of binary fuzzy codes, and the bounds of a Hamming distance of binary fuzzy codes for p=1/r, where r3, and rZ+, are determined. Finding Hamming distance of binary fuzzy codes is used for decoding sent messages on a BSC.

Keywords: classical codes, binary fuzzy codes, fuzzy space and Hamming distance

References(15)

[1]

L. A. Zadeh, Fuzzy sets, Inf. Control, vol. 8, no. 3, pp. 338–353, 1965.

[2]

C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., vol. 27, no. 3, pp. 379–423, 1948.

[3]
C. E. Shannon, Certain results in coding theory for noisy channels, Inf. Control, vol. 1, no. 1, pp. 6–25, 1957.
DOI
[4]

R. W. Hamming, Error detecting and error correcting codes, Bell Syst. Tech. J., vol. 29, no. 2, pp. 147–160, 1950.

[5]

R. C. Bose and D. K. Ray-Chaudhuri, On a class of error correcting binary group codes, Inf. Control, vol. 3, no. 1, pp. 68–79, 1960.

[6]

D. E. Muller, Application of Boolean algebra to switching circuit design and to error detection, Trans. I. R. E. Prof. Group Electron. Comput., vol. EC-3, no. 3, pp. 6–12, 1954.

[7]
I. S. Reed, A class of multiple-error-correcting codes and the decoding scheme, Technical Report, Massachusetts Inst of Tech Lexington Lincoln Lab AD0025814, Lexington, MA, USA, 1953.
[8]

I. S. Reed and G. Solomon, Polynomial codes over certain finite fields, J. Soc. Indust. Appl. Math., vol. 8, no. 2, pp. 300–304, 1960.

[9]

P. A. von Kaenel, Fuzzy codes and distance properties, Fuzzy Sets Syst., vol. 8, no. 2, pp. 199–204, 1982.

[10]
S. Ling and C. Xing, Coding Theory: A First Course. Cambridge, UK: Cambridge University Press, 2004.
DOI
[11]

S. A. Tsafack, S. Ndjeya, L. Strüngmann, and C. Lele, Fuzzy linear codes, Fuzzy Information and Engineering, vol. 10, no. 4, pp. 418–434, 2018.

[12]

B. Amudhambigai and A. Neeraja, A new view on fuzzy codes and its application, Jordan J. Math. Stat., vol. 12, no. 4, pp. 455–471, 2019.

[13]

A. Ozkan and E. M. Ozkan, A different approach to coding theory, J. Appl. Sci., vol. 2, no. 11, pp. 1032–1033, 2002.

[14]

P. Lubczonok, Fuzzy vector spaces, Fuzzy Sets Syst., vol. 38, no. 3, pp. 329–343, 1990.

[15]

R. Lowen, Convex fuzzy sets, Fuzzy Sets Syst., vol. 3, no. 3, pp. 291–310, 1980.

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Received: 13 September 2022
Revised: 20 October 2022
Accepted: 29 October 2022
Published: 06 April 2023
Issue date: March 2023

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© The Author(s) 2023. Published by Tsinghua University Press.

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This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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