[[Linear equivalence of codes]]
# MacWilliams theorem

Let $\varphi : \mathcal{C} \to \mathcal{C}'$ be a linear isomorphism between [[Linear code|$[n,k]$ codes]] $\mathcal{C}, \mathcal{C}'$.
Then if the [[generalized Hamming weight]] $\wt_{t}(\mathcal{D}) = \wt_{t}(\varphi(\mathcal{D}))$ for every $t$-dimensional linear subcode $\mathcal{D} \leq \mathcal{C}$,
then $\varphi$ is a [[Linear equivalence of codes]].[^2011]

  [^2011]: 2011\. [[Sources/@liuEquivalenceLinearCodes2011|On the equivalence of linear codes]]

> [!missing]- Proof
> #missing/proof

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