Can someone prove the following problem?
Let $C \subseteq \mathbb{F}_2^n$ be a linear code and let $$C^{\bot} = \{y \in \mathbb{F}_2^n \mid \langle x,y \rangle \mbox{ for all} \; x \in C\},$$ be the dual code of $C.$ Suppose the minimal (hamming) distance of codes in $C$ is $k+1$
Show that for any subset $S$ of $l \leq k$ coordinates every $(0,1)$-vector of lenght $l$ appears as a projection of $C^{\bot}$ onto $S$ the same number of times.
Hint: If those $\ell$ columns in a generator matrix of $C^\perp$ (= a parity check matrix of $C$) were linearly dependent, what would it say about the minimum distance of $C$?