91Ó°ÊÓ

Suppose we encode the four vectors in \(\mathbb{Z}_{2}^{2}\) by repeating the vector twice. Thus, we have $$\begin{array}{l} {[0,0] \rightarrow[0,0,0,0]} \\ {[0,1] \rightarrow[0,1,0,1]} \\ {[1,0] \rightarrow[1,0,1,0]} \\ {[1,1] \rightarrow[1,1,1,1]} \end{array}$$ Show that this code is not error-correcting.

Suppose we encode the binary digits 0 and 1 by repeating each digit five times. Thus, $$\begin{array}{l} 0 \rightarrow[0,0,0,0,0] \\ 1 \rightarrow[1,1,1,1,1] \end{array}$$ Show that this code can correct double errors.

When the (7,4) Hamming code of Example 3.71 is used suppose the messages \(\mathbf{c}^{\prime}\) are received. Apply the standard parity check matrix to \(\mathbf{c}^{\prime}\) to determine.whether an error has occurred and correctly decode \(\mathrm{c}^{\prime}\) to recover the original message vector \(\mathbf{x}\). $$\mathbf{c}^{\prime}=\left[\begin{array}{lllllll} 0 & 0 & 1 & 1 & 1 & 1 & 0 \end{array}\right]^{T}$$

Define a code \(\mathbb{Z}_{2}^{2} \rightarrow \mathbb{Z}_{2}^{5}\) using the standard generator matrix $$G=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \\ 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{array}\right]$$ (a) List all four code words. (b) Find the associated standard parity check matrix for this code. Is this code (single) error-correcting?

Define a code \(\mathbb{Z}_{2}^{3} \rightarrow \mathbb{Z}_{2}^{6}\) using the standard generator matrix $$G=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right]$$ (a) List all eight code words. (b) Find the associated standard parity check matrix for this code. Is this code (single) error-correcting?