# Is the union between two cyclic codes always cyclic

I confuse that is it always necessary for the union of two cyclic codes is a cyclic code?

I try to prove that it is not necessarily the union of two cyclic codes is a cyclic code.

This is my counterexample:

Let $$C_1$$ and $$C_2$$ be two cyclic codes over $$F_2$$.

$$C_1 = \{000, 101, 011, 110\}$$

$$C_2 = \{000, 111\}$$

Note that $$C_1$$ and $$C_2$$ are cylic codes. Hence,

$$C_1 \cup C_2=\{000, 101, 011, 110, 111\}$$

However $$111+011=100\notin C_1\cup C_2$$. Hence $$C_1 \cup C_2$$ is not a linear code. Since it is not a linear code then it is not a cyclic code.

Is the counterexample is correct?

One definition is that a code is called a cyclic code if the cyclic shifts of every codeword in the code are also codewords in the code. With this definition, $$\mathcal C_1 \cup \mathcal C_2$$ is a cyclic code.
A more restrictive definition includes the requirement that the code be linear as well as satisfying the cyclic shifts property mentioned above. With this restrictive definition, it is not true that $$\mathcal C_1 \cup \mathcal C_2$$ is a (linear) cyclic code whenever $$\mathcal C_1$$ and $$\mathcal C_2$$ are (linear) cyclic codes. In this case, it can be shown that $$\mathcal C_1 + \mathcal C_2$$ is a (linear) cyclic code whenever $$\mathcal C_1$$ and $$\mathcal C_2$$ are (linear) cyclic codes. Here $$\mathcal C_1 + \mathcal C_2$$ is defined as $$\mathcal C_1 + \mathcal C_2 = \left\{\mathbf c_1 + \mathbf c_2\colon \mathbf c_1 \in \mathcal C_1, \mathbf c_2 \in \mathcal C_2\right\}$$