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how to generate a galois field extension array without using any built-in functions?

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hello,
i want to generate a GF(2.^m) array for a given polynomial such that: 1. first the polynomial irreducibility is checked,
2.then if it is primitive or not and if yes
3. then all possible polynomials in vector form should be listed in a matrix
i cant use built-in matlab functions.

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