I am interested in working with Galois field in MATLAB. Can any one explain for me with example if it possible the following definition:
I know $GF(2)=\{0,1\}$, but what does $GF(2^m)$ mean? If $m=3$, can any one give me the table of addition and multiplication, and if it possible some MATLAB function, please?
Check out the function gf in Matlab. It convert numbers from range $0$ to $2^m-1$ to an object in $GF(2^m)$
$GF(2^m)$ is a finite field with $2^m$ elements. It can be generated using a primitive root from a primitive polynomial.
This Mathwork document teach us how to create multiplication table and addition table. Read the section on arithmetic on galois field.
Edit:
primitive polynomial: $1+x+x^2$.
The elements are $1, \alpha, 1+\alpha, 0$.
Multiplication of $1$ and $0$ are trivial.
$$\alpha(\alpha) = \alpha^2 = \alpha +1$$
$$\alpha(\alpha+1)=\alpha^2+\alpha=1$$
$$(\alpha+1)(\alpha+1)=\alpha^2+1=\alpha$$