How was the AES MixColumns matrix constructed? How was the constant matrix in AES MixColumns constructed? 

\begin{bmatrix}
02 &amp; 03 &amp; 01 &amp; 01 \\
01 &amp; 02 &amp; 03 &amp; 01 \\
01 &amp; 01 &amp; 02 &amp; 03 \\
03 &amp; 01 &amp; 01 &amp; 02 \\
\end{bmatrix}

How can I construct in $GF(2^4)$ an invertible matrix for the MixColumns step?
Any full rank $n \times n$ circulant matrix $A$ with all entries nonzero generates an MDS code. The entries here were picked for efficient fast multiplication, as far as I can recall.

Do refer to the book suggested by @SEJPM for more.