Updates on Properties of Hamming Distance of Binary Fuzzy Codes over Fuzzy Vector Spaces and Their Application in Decoding Process

Abstract

In this paper, we used the definition of fuzzy vector spaces and fuzzy subsets in combination in order to define fuzzy codes over a fuzzy vector space. We found that the combined definitions did not satisfy the definition of fuzzy vector spaces over a Galois field F2, and gave the conditions in which the combined definitions hold the definition of fuzzy vector spaces in order to define binary fuzzy codes. By defining binary fuzzy codes over fuzzy vector space in relation to the probability of binary symmetric channels (BSC) sending a codeword incorrectly and the weight of error pattern between codewords, we updated properties of Hamming distance of binary fuzzy codes over fuzzy vector space. From the updated properties, we found that the properties of Hamming distance stated over a vector space also satisfy in binary fuzzy codes defined over a fuzzy vector space. Furthermore, we found some interesting results on the decoding of fuzzy codewords sent over a BSC using minimum non-zero Hamming distance of binary fuzzy codes.

Keywords

• classical codes
• binary fuzzy codes
• fuzzy vector space
• hamming distance
• decoding fuzzy codes