Dimensions
Task number: 4032
For a linear code \( C \) of length \( n \) over the field \( \mathbb Z_q \) prove: \(\dim C + \dim C^{\bot} = n\).
Solution
If we build a matrix \(\mathbf A\) from the basis of the code \(C\) then \(\dim(C)={\rm rank}(\mathbf A)\).
The vectors of the dual code \(C^{\bot}\) form the kernel of the matrix \( \mathbf A \), this follows directly from the definitions of both terms.
Now just use the relation: \({\rm rank}(\mathbf A) + \dim(\ker(\mathbf A))=n\).
