Situations of the result of Hamming Decoding

Using the resultant $$n − k$$ = 3-bit syndrome $$\mathbf{s}$$, the Hamming decoder decides if it thinks there are any bit errors in $$\hat{\mathbf{y}}$$:
- If the syndrome is $$\mathbf{s} = \begin{array}{ccc}[ 0 &0 &0]\end{array}^{T}$$ then the Hamming decoder thinks there are no bit errors in $$\hat{\mathbf{y}}$$ (it may be wrong though). In this case, it outputs $$\hat{\mathbf{x}} = \begin{array}{cccc}[ \hat{y}_{3} &\hat{y}_{5} &\hat{y}_{6} &\hat{y}_{7} ]\end{array}^{T}$$ since $$y_{3} = x_{1}$$, $$y_{5} = x_{2}$$, $$y_{6} = x_{3}$$ and $$y_{7} = x_{4}$$ in $$\mathbf{G}$$.
- If the syndrome $$\mathbf{s}$$ is not equal to $$\begin{array}{ccc}[ 0 &0 &0]\end{array}^{T}$$ then its 3-bit number is converted into a decimal number $$i ∈ [1, 7]$$. In this case, the Hamming decoder thinks that the ith bit in $$\hat{\mathbf{y}}$$ has been flipped by a bit error (it may be wrong though). The Hamming decoder flips the $$i$$th bit in $$\hat{\mathbf{y}}$$ before outputting $$\hat{\mathbf{x}}=\left[\begin{array}{llll}\hat{y}_3 & \hat{y}_5 & \hat{y}_6 & \hat{y}_7\end{array}\right]^T$$. If there are multiple bit errors in the received codeword $$\hat{\mathbf{y}}$$, the syndrome $$\mathbf{s}$$ identifies which bit of $$\hat{\mathbf{y}}$$ can be toggled to give the legitimate permutation of $$\mathbf{y}$$ that is most similar.

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