Sum of two cyclic codes is a cyclic code.

Question

My text made a small comment about cyclic codes:

The sum of two cyclic codes $C_1$ and $C_2$, defined by:

$C_1+C_2=\{c_1+c_2: c_1\in C_1, c_2\in C_2 \}$

The sum of two cyclic codes is also a cyclic code.

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I am unsure of how this proof would start, so I am looking for suggestions (my text does not provide a proof of this fact, so I am trying to do it for myself).

Answer 1

Hint/plan of attack: Let $S$ stand for the operation of cyclically shifting a codeword by one position.

1. Show that $S(c_1+c_2)=S(c_1)+S(c_2)$ for all words $c_1$ and $c_2$.
2. Show that if $C_1$ and $C_2$ have the property that $S(C_i)\subseteq C_i$ for $i=1,2$, then also $C_1+C_2$ has this property.