We have two definitions for MDS (Maximum Distance Separable) Matrix:
First definition: A matrix $M$ of order $n$ is an MDS matrix if and only if every sub-matrix of $M$ is non-singular.
Second definition: A matrix $M_{n\times n}$ is MDS if and only if
$$ Y_{n\times 1}=M_{n \times n}\, X_{n \times 1} \Longrightarrow \mathop{\rm min}_{X\neq 0}(W(Y)+W(X))=n+1 $$
where $X={[x_0,x_1,\cdots , x_{n-1}]}^T$ and $Y={[y_0,y_1,\cdots , y_{n-1}]}^T$ are vectors in an arbitrary field and $W(X)$ is the number of non-zero elements of $X$.
I have two questions:
Is it possible to make an example a matrix of order at least $3$ and by second definition proof that is an MDS matrix and my next question is that Is there a suitable book or article about MDS matrix.
In fact, I am researching about new simple definition or conditions such that a matrix be MDS.
I would be appreciate for any suggestions.