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consider the field F_2^4, defined by using polynomial representation with the irreducible polynomial f(x) = x^4 + x + 1.

Given element g = (0010) as a generator for the field. How are the powers of g generated? What are the steps to generate the binary values of g0 to g15 as follows?

g0 = (0001) g1 = (0010) g2 = (0100) g3 = (1000) g4 = (0011) g5 = (0110)

g6 = (1100) g7 = (1011) g8 = (0101) g9 = (1010) g10 = (0111) g11 = (1110)

g12 = (1111) g13 = (1101) g14 = (1001) g15 = (0001)

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  • $\begingroup$ The sequence is a notation for the successive polynomial powers of $x$ modulo $x^4+x+1$, over $F_2$. Here $abcd$ indicates $ax^3+bx^2+cx+d$. Does this answer your question? $\endgroup$ Jul 6, 2014 at 5:12
  • $\begingroup$ How are the powers of g generated? If only, x^4 + x + 1 over field F_2^4, and element g = (0010) as the generator for the field is assumed? What are the steps to generate the binary values of g0 to g15? $\endgroup$
    – Ursa Major
    Jul 6, 2014 at 5:16
  • $\begingroup$ @UrsaMajor: This kind of question occurs frequently enough. I wrote a community wiki Question/Answer for the purposes of directing people to it. What you refer to as g0, g1,...,g15 are what I denote by $\gamma^i, i=0,1,\ldots,15,$ in that answer. The others are right. This is not research level math. If you need more explanations, @-ping me at that Math.SE-question or (possibly better) ask it as question at Math.SE. $\endgroup$ Jul 6, 2014 at 16:44
  • $\begingroup$ @JyrkiLahtonen, yes, I think you got me the concept. How about how to write a pseudo code for the computer? The concept you did is good, but how to instruct the computer to generate automatically? Are you well versed with coding? Could you write down the steps for the computer to generate the binary values of g0 to g15 given the polynomial? $\endgroup$
    – Ursa Major
    Jul 6, 2014 at 21:35

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