Self-dual code
Task number: 4035
Construct a self-dual code above \( \mathbb Z_2 \) (i.e. \( C = C^{\bot} \)) of length at least \( 2 \).
Solution
For each word to be dual to itself, it must have an even number of ones.
Now it is enough for the ones from any two words in the code to match in an even number of positions.
Suitable codes are therefore, for example, a repetitive code of length \( 2 \), or more generally codes that arise from the (complete) code \( Z_2^n \) over \( \mathbb Z_2 \) by doubling all symbols.
Note that if we duplicate the symbols in the \( C \) code, which would not be complete, \( \dim(C) < n \) would apply. As \(\dim(C)+\dim(C^\bot)=2n\), we get that \( C^\bot \) has larger dimension than \( C \), and therefore they do not match.
