I am a bachelor student who just started studying coding theory and I came across the following example in "Generalized Hamming weights for almost affine codes" by Johnsen and Verdure, page 4 (https://arxiv.org/abs/1601.01504):
We will use a running example throughout this paper. It is the almost affine code C
′
in [14, Example 5]. It is a code of length 3 and dimension 2 on the alphabet F = {0, 1, 2, 3}. Its set
of codewords is
$$
\begin{matrix}
000 & 011 & 022 & 033 \\
101 & 112 & 123 & 130 \\
202 & 213 & 220 & 231 \\
303 & 310 & 321 & 332 \\
\end{matrix}
$$
[…] This is an example of an almost affine code which is not
equivalent to a linear code, and not even to a multilinear code.
I understand why the code is almost affine. However, to me, it seems like this is a linear code, since all the codewords can be expressed as linear combinations of 101 and 011. What mistake am I making? Also, how would I check that it is not equivalent to a multilinear code? Thank you in advance.
Best Answer
When people say a $C$ linear code, they mean that $C$ is linear subspace of a vector space. It is true that $C'$ is a sub-group of the group $(\mathbb Z/4\mathbb Z)^3$, generated by the two group elements $(1,0,1)$ and $(0,1,1)$, but since it is not a sub-space of a vector space, we cannot call $C'$ linear. Note that $\mathbb Z/4\mathbb Z$ is not a field.
I do not know about how to prove it is not equivalent to a (multi)linear code, perhaps someone else can answer about that.