Question about block erasure codes I have a question about linear block erasure codes that are described in this paper.
I briefly describe the idea behind the linear erasure codes and then I ask my question.

Given a set $d=\langle x_i \in GF(2^q)|i=1,\cdots,k\rangle$ of data packets, the general idea of (systematic) erasure codes is to generate $n-k$ extra packets $e=\langle y_j \in GF(2^q)|j=1,\cdots,n-k\rangle$, such that given any $k$ packets out of the set $\{x_1,\cdots,x_k,y_1,\cdots,y_{n-k}\}$ one is able to decode the original set $d$.

To generate the extra packets, we consider an $(n-k) \times n$ generator matrix $G'$ and compute $e=G'd$.

The paper claims that if the rows of the generator matrix are selected from a Vandermonde matrix:

$$
V=\begin{bmatrix}
1 &amp; \alpha_1 &amp; \alpha_1^2 &amp; \dots &amp; \alpha_1^{k-1}\\
1 &amp; \alpha_2 &amp; \alpha_2^2 &amp; \dots &amp; \alpha_2^{k-1}\\
1 &amp; \alpha_3 &amp; \alpha_3^2 &amp; \dots &amp; \alpha_3^{k-1}\\
\vdots &amp; \vdots &amp; \vdots &amp; \ddots &amp;\vdots \\
1 &amp; \alpha_m &amp; \alpha_m^2 &amp; \dots &amp; \alpha_m^{k-1}
\end{bmatrix}
$$
where $\alpha_i \in GF(2^q)$
then the computed vector $e$ has the desired property.

Now my question: Assume $n-k=2$, i.e., any loss of 2 packets can be recovered.
Let,
$$
G'=\begin{bmatrix}
1 &amp; \alpha_1 &amp; \alpha_1^2 &amp; \dots &amp; \alpha_1^{k-1}\\
1 &amp; \alpha_2 &amp; \alpha_2^2 &amp; \dots &amp; \alpha_2^{k-1}
\end{bmatrix}
$$

By definition of the erasure codes, the submatrix consisting of any two columns of the matrix $G'$ must have rank 2. But we know that for $\alpha_1, \alpha_2 \in GF(2^q)$ and $\alpha_1 \neq \alpha_2$, we may have $\alpha_1^s = \alpha_2^s$ for some $s$.
It means that the matrix 

$$
\begin{bmatrix}
1 &amp; \alpha_1^s \\
1 &amp; \alpha_2^s
\end{bmatrix}
$$
has rank 1 rather than 2. So, it means that if packets $x_1$ and $x_{s+1}$ are lost, they cannot be recovered using the set $e$.
Generally, the fact that an $m\times m$ Vandermonde matrix is non-singular,
does not mean that any submatrix of it has full rank, does it?

Did I misunderstood the concept or is it the fault of the paper?
Ok, I found the answer.

The answer is yes, singular submatrices may happen in a Vandermonde matrix. Look at this paper for an approach to avoid singular submatrices while creating a generator matrix:

http://oatao.univ-toulouse.fr/2176/1/Lacan_2176.pdf