The problem says: Supose that H is a control matrix of a $[n,k]_q$-code C and let's consider $x\in\mathbb{F}_q^n$ wich $s(x)=xH^t=v\in\mathbb{F}_q^{n-k}$.
If $y\in x+C$ and $w(y)=t$ then show that $v$ is a linear combination of $t$ H's columns.
I'm quite stuck on this problem, like I need some clue.
HM.
Let $y\in x+C$. Then $y=x+c$ for some $c\in C$. Thus $yH^t = (x+c)H^t= xH^t + cH^t = xH^t =v$. If the weight of $y$ is $t$, then $v=yH^t$ is indeed a linear combination of $t$ columns of $H$ (just by looking at the matrix-vector multiplication).