(4,3) even parity code is linear, odd parity is not

A block code is linear if (1) the all-zero word is a codeword and (2) the modulo-2 sum of any two codewords is also a codeword (closure). For binary codes, (2) implies (1) since $\mathbf{c}\oplus\mathbf{c}=\mathbf{0}$.

A (4,3) parity code takes 3 message bits $m_1m_2m_3$ and adds one parity bit $p$.

Even parity code is linear

$p=m_1\oplus m_2\oplus m_3$, so every codeword has an even number of 1s:

General proof: let $\mathbf{c}_1,\mathbf{c}_2$ be codewords, with weights $w_1,w_2$ both even. Then

$$w(\mathbf{c}_1\oplus\mathbf{c}_2)=w_1+w_2-2\,w(\mathbf{c}_1\cdot\mathbf{c}_2)$$

where $\mathbf{c}_1\cdot\mathbf{c}_2$ counts positions where both are 1. Even + even − even = even, so the sum is again an even-weight 4-bit word, i.e. a codeword. Also $0000$ is a codeword.

Equivalently, the code can be generated by a matrix:

$$G=\begin{bmatrix}1&0&0&1\\0&1&0&1\\0&0&1&1\end{bmatrix},\qquad \mathbf{c}=\mathbf{m}G$$

and $\mathbf{m}_1G\oplus\mathbf{m}_2G=(\mathbf{m}_1\oplus\mathbf{m}_2)G$, which is a codeword.

Example: $0011\oplus0101=0110$ (codeword), $1001\oplus1111=0110$ (codeword). Hence the (4,3) even parity code is a linear block code.

Odd parity code is not linear

$p=m_1\oplus m_2\oplus m_3\oplus1$, so every codeword has an odd number of 1s:

1. The all-zero word $0000$ has even weight, so it is not a codeword.
2. Closure fails: for odd $w_1,w_2$, $w_1+w_2-2(\cdot)$ is even, so the sum of two codewords always has even weight and is never a codeword. Example: $0001\oplus0010=0011$, weight 2, not in the code.

Hence the (4,3) odd parity code is not a linear block code.