Proof that a linear code union with another linear code is a linear code

coding-theorylinear algebra

  1. Let C be a binary linear code: Proof or disproof $C∪(C+x) $ is a linear code.

For this task my solution idea is:

$c_1, c_2 \in C $ therefore (because $C$ is linear) $c_1 + c_2 \in C$

($c_1 + x) + (c_2 + x) = c_1 + c_2 +2x = c_1+c_2$

Because $C$ is linear code and all calculations are modulo 2 we do not need to show scalar multiplication. (Is this right?)

Therefore $C∪(C+x)$ is linear.

  1. Let $C, D$ be binary linear codes. Proof or disproof that $C \cup D$ is also linear code.

For this task I have no idea (does task 1. help?), any help is welcome.

Best Answer

The first part is right, and there is another question that asks the same, "answered" in the comments.

The second part is false. Consider $C=\langle (1,0)\rangle$ and $D=\langle (0,1)\rangle$. It is clear that $(1,0)+(0,1)=(1,1)\notin C\cup D$, so it is not closed under addition.

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