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.

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 $r\in {\mathbb{Z}}^{+}$, are determined. Finding Hamming distance of binary fuzzy codes is used for decoding sent messages on a BSC.