Construction of Goppa codes

We construct a Goppa code of length 31 over GF(2) with generator polynomial .

> q := 2^5;
> K<w> := FiniteField(q);
> P<z> := PolynomialRing(K);
> G := z^3 + z + 1;
> L := [w^i : i in [0 .. q - 2]];
> C := GoppaCode(L, G);
> C;
[31, 16, 7] Goppa code (r = 3) over GF(2)
Generator matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 0 0 0 0 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1]
[0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1]
[0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 1 1 1 1]

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