The linearity of the extended code

Question

If we add a last digit to the code $C$ of length $n$, we obtain a new code called extended code. My question is:

If the code $C$ is linear, can we prove that the extended code $C'$ is linear too?

Answer 1

If you add an extra $0$ to every code word, we will have a linear code, with the same minimum distance. Just adding any digit won't give a linear code in general. I suppose if you start with a basis for the code you can add an arbitrary digit to each basis element and then extend linearly to the other code words, and get better distance properties, in principle.

Answer 2

A typical extension is to add an overall parity check symbol to $C$, meaning that after the extension, the sum of the entries of any codeword in $C'$ equals $0$. This extended code $C'$ is indeed linear (since the extension rule is linear).

If you just add random symbols, in general $C'$ is not linear.

In general, the extended code C′ is linear if and only if the map $C\to\mathbb F_ q$, mapping a codeword to its extension symbol is linear.