In this question, we are going to construct a $(7,3)$ linear block code by extending a $(6,3)$ linear block

code. The generator matrix of the $(6,3)$ code is

G $=$

$$

$$

$100110$

$010101$

$001011$

$$

$$

The $(7,3)$ code is extended from the $(6,3)$ code by appending one extra bit to into the codeword such

that the Hamming weight of the resulting codeword is even. For example, for an input information sequence of $[111],$ the $(6,3)$ code produces an codeword of $[111000]$ which is then extended by appending

an extra bit $1$ at the end to form the final codeword of $[1110001].$ The reason for appending a bit $1$

$($instead of bit $0)$ is to make the Hamming weight of the resulting codeword in this example be an

even number.

Another way to describe the $(7,3)$ code would be that it consists of a $(6,3)$ code with generator matrix

G followed by another single parity check code with even parity