Studies on Binary Fuzzy Codes, and Binary Fuzzy Linear Codes Over Finite Fuzzy Vector Spaces

Manmekto, Mezgebu

dc.contributor.author	Manmekto, Mezgebu
dc.date.accessioned	2024-07-24T10:56:39Z
dc.date.available	2024-07-24T10:56:39Z
dc.date.issued	2024-07
dc.identifier.uri	http://ir.bdu.edu.et/handle/123456789/15899
dc.description.abstract	In this dissertation, we discuss binary fuzzy codes over an n-dimensional vector space F n 2 . We accomplish this by redefining fuzzy subsets over F n 2 in a new manner using the formula for the reliability of the channel to send messages correctly over noisy binary symmetric channels (BSC). We use the weight of error patterns e between a received word and the possible sent codewords to define fuzzy words over F n 2 and use it to define binary fuzzy codes. The binary fuzzy code is defined after redefining the fuzzy subset. Additionally, we add properties of the fuzzy Hamming distance of binary fuzzy codes, and we determine the bounds of the fuzzy Hamming distance of binary fuzzy codes for any p ∈ (0, 1 2 ). Finding the fuzzy Hamming distance of binary fuzzy codes is useful for minimum distance decoding, and we checked the decoding algorithms of minimum distance decoding of classical codes for fuzzy codes. The algorithms also work for the decoding of binary fuzzy codes in general. We also explore the fuzzy Euclidean distance of fuzzy codes and realize that the fuzzy Euclidean distance can be used for decoding binary fuzzy codes. However, the fuzzy Euclidean distance is more complex in computation than the fuzzy Hamming distance. Therefore, we recommend using the fuzzy Hamming distance for minimum distance decoding unless it is needed for a special purpose. We also use the definition of fuzzy vector spaces by Lubczonok and the redefined fuzzy subsets and find that the combined definitions do not satisfy the definition of fuzzy vector spaces f n 2 over a Galois field F2 . We provide the conditions under which the combined definitions hold the definition of f n 2 to define binary fuzzy codes over it. Binary fuzzy codes over f n 2 are defined in relation to the probability p of the BSC sending a codeword incorrectly and the weight of error pattern e between codewords. We update the fuzzy distance properties and the decoding of binary fuzzy codes over vii f n 2 . From the updated properties, we realize that the distance properties and decoding algorithms of binary fuzzy codes over F n 2 are satisfied for binary fuzzy codes defined over f n 2 . However, studying binary fuzzy codes provides the advantage of determin ing the fuzzy dimension of the codes and theoretically thinking about representing the ignored errors in coding theory. In the dissertation, we also explore the concept of binary convex fuzzy vector spaces and binary convex fuzzy linear subspaces over F n 2 by formalizing their definition, and we found interesting results on their properties. By establishing relevant properties, we studied binary fuzzy vector spaces and convex fuzzy sets. We accomplish this by redefining fuzzy subsets over F n 2 to make the concepts applicable in natural and lin guistic communication systems. Furthermore, we realized that the properties discussed for binary fuzzy vector spaces are applied to binary convex fuzzy vector spaces. We also defined binary convex fuzzy codes over binary convex fuzzy vector space and pre sented the properties we have seen for binary fuzzy codes above. However, we realized that the properties found for binary fuzzy codes also apply to binary fuzzy codes. The minimum distance decoding of binary fuzzy codes is applied to binary convex fuzzy codes. Our examination of fuzzy distances revealed that distinct fuzzy codewords with the minimum Hamming distance also have the minimum Euclidean distance within a given binary fuzzy code. Furthermore, our study revealed that the Hamming distance is influenced by two parameters, p, and e, whereas the Euclidean distance depends on these parameters as well as the code length n. Finally, we studied the properties of binary fuzzy subspaces over F n 2 to apply the concepts in the study of binary linear fuzzy codes. The redefined fuzzy subset men tioned above does not fit the definition of fuzzy subspaces. Therefore, we redefined the fuzzy subset once again in the form of a piecewise function by specifying the condi tions. However, this definition only holds for vectors selected from the two-dimensional vector space F 2 2 . We outlined the conditions under which the definition is valid and ex plained its failure for lengths greater than or equal to 3, clarifying that the redefined fuzzy subset is a fuzzy subspace. Additionally, we delved into the properties of fuzzy viii subspaces to define and analyze binary linear fuzzy codes. Through our exploration of binary linear fuzzy codes, we identified the properties of codes with a length of 2. Lastly, we defined binary linear fuzzy codes over F n 2 and f n 2 for n ≥ 3, with the properties and releted concepts being subjects for our future work	en_US
dc.language.iso	en_US	en_US
dc.subject	Mathematics	en_US
dc.title	Studies on Binary Fuzzy Codes, and Binary Fuzzy Linear Codes Over Finite Fuzzy Vector Spaces	en_US
dc.type	Dissartation	en_US