Binary Fuzzy Codes and Some Properties of Hamming Distance of 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.