Given a linear code $C$, then the subset $D\subset C$ with $D:=\{v\in C : w(v)=2k \textrm{ for }k\in \mathbb N \}$ is a linear code

Question

I found the problem above in one of the previpous year exams for my discrete mathematics course but I think it might be wrong.

I've seen similar problems but they all add the hypothesis of $C$ being binary. This way, when analyzing the different cases for the addition of two elements of $D$ I get that $w(x+y)=w(x)+w(y)-2\{i\in \{1,...,n\} : x_i=y_i=1\}$ . This way, I can make sure the weight of the sum of two elements in $D$ is even and therefore the sum is an internal operation in $D$. That added to the fact that the element $0$ has an even weight ensures that $D$ is a vector space and therefore a lineal code.

However, when analyzing a field different to $\mathbb F_2^n$ I fail to find that $w(x,y)$ has to be even.

In fact, couldn't I just use $x=(1,2,0), y=(2,2,0) \in \mathbb F_3^3 $ such that $(x+y)=(0,1,0)$ where $w(x)=2, w(y)=2$ but $w(x+y)=1$ as a counter example?

I'd like someone to verify if the property is true or if my professor made a mistake in the exam.

I'd also apreciate it if someone checked the my reasoning on the binary part.

Thanks in advance

Answer 1

You are correct; the statement you are asked to prove is true for linear binary codes but not for codes over larger fields (including extensions of $\mathbb F_2$). If you are asked the same question on an exam, I would suggest giving the proof for linear binary codes, and your counterexample to show the professor that the result to be proved is not true in general, and needs additional constraints to make it valid.