Suppose $C$ is a linear code, with elements $\textbf{c}$ and $\textbf{v}$ is a code word not in $C$, but it is an element of We form the coset of $C$, given by $$ C + \textbf{v} = \{\textbf{c} + \textbf{v} \ : \ \textbf{c} \in C, \textbf{v} \not \in X\}. $$ I want to prove that $C \cup C + \textbf{v}$. is also a linear code, but am struggling a bit.
I considered breaking this into cases by considering a partition $C \cup (C + \textbf{v})$, and then establishing closure for each of the partitions. But, there seems to be a lapse in information about $\textbf{v}$, save for the fact that it's a field element and thus clearly is closed under the group operation.
Any comments or insights would be greatly appreciated.
That's true iff $C$ is a binary linear code, that is it's linear over $\Bbb F_2$, the field of two elements.
The difficult case is showing the sum of two elements of $C+v$ lies in $C\cup(C+v)$. If those elements are $c_1+v$ and $c_2+v$ then they add to $$(c_1+v)+(c_2+v)=c_1+c_2+2v=c_1+c_2\in C$$ as $2=0$ within $\Bbb F_2$.