Referring to elements of a polynomial ring as integers.

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First, you can transform a decimal number into a string representing its binary expansion as follows:

sage: n = 14
sage: b = n.bits()
sage: b
[0, 1, 1, 1]

Then you can get the indices of the letters by enumerating this iterator:

sage: list(enumerate(b))
[(0, 0), (1, 1), (2, 1), (3, 1)]

Putting everything together, you can construct your element of F:

sage: sum(a^i for i,j in enumerate(n.bits()) if j)
a^3 + a^2 + a

Note that you can go the other way easily:

sage: c = a^3 + a^2 + a
sage: c.integer_representation()
14

Regarding the index, since F^* is multiplicatively cyclic with a as a generator, you can define and use the following dictionary:

sage: index = {a^i:i for i in range(15)}
sage: index[a^3 + a^2 + a]
11

Now, regarding polynomials, you can first construct the integer polynomial to represent them:

sage: D.<X> = PolynomialRing(ZZ)

Then, you can construct your example polynomial as follows:

sage: P = sum((11-i)*X^i for i in range(11))
sage: P
X^10 + 2*X^9 + 3*X^8 + 4*X^7 + 5*X^6 + 6*X^5 + 7*X^4 + 8*X^3 + 9*X^2 + 10*X + 11

A nice way to access coefficients of a polynomial is by using a dictionary that maps powers to coefficients, so that at the end, you get:

sage: sum([sum(a^i for i,j in enumerate(ZZ(coeff).bits()) if j) * x^power for power,coeff in P.dict().items()])
x^10 + a*x^9 + (a + 1)*x^8 + a^2*x^7 + (a^2 + 1)*x^6 + (a^2 + a)*x^5 + (a^2 + a + 1)*x^4 + a^3*x^3 + (a^3 + 1)*x^2 + (a^3 + a)*x + a^3 + a + 1

If you want the index-ed form, as an integer polynomial:

sage: sum(index[sum(a^i for i,j in enumerate(ZZ(coeff).bits()) if j)] * X^power for power,coeff in P.dict().items())
X^9 + 4*X^8 + 2*X^7 + 8*X^6 + 5*X^5 + 10*X^4 + 3*X^3 + 14*X^2 + 9*X + 7

All that said, note that Reed-Solomon codes ar already implemented in Sage https://doc.sagemath.org/html/en/refe...