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: \({\rm rank}(\mathbf A) + \dim(\ker(\mathbf A))=n\).

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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