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

Mezgebu Manmekto Gereme; Jejaw Demamu; Berhanu Assaye Alaba

doi:10.26599/FIE.2023.9270003

| Sign up

PDF (1.3 MB)

Cite

EndNote(RIS) BibTeX

Collect

Submit Manuscript

Article | Open Access

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

Mezgebu Manmekto Gereme^{¹^,²}

(), Jejaw Demamu^{¹^,²}, Berhanu Assaye Alaba^¹

1Department of Mathematics, Bahir Dar University, Bahir Dar 6000, Ethiopia

2Department of Mathematics, Debark University, Debark 6200, Ethiopia

Show Author Information

Abstract

In this paper, we discussed binary fuzzy codes over a vector space $F_{2}^{n}$ 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 $F_{2}^{n}$ , 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 $r ⩾ 3$ , and $r \in Z^{+}$ , 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

[1]

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

Crossref Google Scholar

[2]

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

Crossref Google Scholar

[3]

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

Crossref

[4]

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

Crossref Google Scholar

[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.

Crossref Google Scholar

[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.

Crossref Google Scholar

[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.

Crossref Google Scholar

[9]

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

Crossref Google Scholar

[10]

S. Ling and C. Xing, Coding Theory: A First Course. Cambridge, UK: Cambridge University Press, 2004.

Crossref

[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.

Crossref Google Scholar

[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.

Google Scholar

[13]

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

Crossref Google Scholar

[14]

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

Crossref Google Scholar

[15]

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

Crossref Google Scholar

Fuzzy Information and Engineering

Volume 15 Issue 1,
March 2023

Pages 26-35

DOI: 10.26599/FIE.2023.9270003

Cite this article:

Gereme MM, Demamu J, Alaba BA. Binary Fuzzy Codes and Some Properties of Hamming Distance of Fuzzy Codes. Fuzzy Information and Engineering, 2023, 15(1): 26-35. https://doi.org/10.26599/FIE.2023.9270003