Erasure Codes with Simplex Locality In here, theorem $1.1$. there is this line 'Since $G$ has full rank it is possible to enlarge $N$ to a set $N^{'}$ ... exactly $k-1$. Note that the enlargement operation ... any of the leaders' in the proof. I don't understand the two sentences. What are these 'corresponding columns' in $G$? Why locality ensures such enlargement operation can be done? !enter image description here

Let me know if you have any doubt with any of the steps:

1. Matrix $G$ is of size $k \times n$; because it's of full rank, its column rank (the dimension of the space spaned by all its columns) is $k$. The "nodes" (indexes of the codeword) correspond to the matrix columns.

2. We pick a set of $\lfloor (k-1)/r \rfloor$ columns as "leaders". The set of "friends" of the "leaders", $N$, comprises less than $k$ columns, so the column rank of this submatrix is less than $k$.

3. It must be possible to pick some more columns ("enlarge $N$") so that the column rank of the enlarged matrix is $k-1$.

4. And it must be possible to do that without using the "leader" columns (why? because the leader columns are a linear combination of the friend columns, so they lie in the same space, so they don't increase the rank).