F is finite field with cardinality q and $F_k[x]=\{f(x)\in F[x] | deg(f(x)) \leq k \}$. We choose $x_1,...,x_n$ elements of F different from each other and for $k \leq n$ we consider a code $C=\{(f(x_1),...,f(x_n)) | f(x) \in F_k[x]\}$. Show that C is linear code and compute parameters.
Is it true that this code has generetor matrix the (nxk) matrix
\begin{equation*} \begin{bmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{k-1}\\ 1 &x_2 & x_2^2 & \cdots & x_2^{k-1}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{k-1} \end{bmatrix} \end{equation*}
Indeed, it's right. Take the information vector $c=(c_0,\ldots,c_{k-1})$ or equivalently the polynomial $f(x) = c_0 + c_1x + \ldots + c_{k-1}x^{k-1}$. Then the encoding gives $$(c_0,\ldots,c_{k-1}) \cdot \left(\begin{array}{cccc} 1 & 1 & \ldots,& 1\\ x_1 & x_2 & \ldots & x_n\\ \vdots & \vdots & \ddots &\vdots\\ x_1^{k-1} & x_2^{k-1} & \ldots & x_n^{k-1} \end{array}\right)\\ = \left( \sum_i c_ix_1^i, \sum_i c_ix_2^i, \ldots, \sum_i c_ix_n^i\right) = ( f(x_1), f(x_2), \ldots ,f(x_n)). $$ All vectors are row vectors and the generator matrix is a $k\times n$ matrix here.