Goppa Code - Function Code

Function Code

The function code (or dual code) with respect to a curve X, a divisor G and the set is constructed as follows.
Let, be a divisor, with the defined as above. We usually denote a Goppa code by C(D,G). We now know all we need to define the Goppa code:

C(D,G) = {(f(P1), ..., f(Pn))|f L(G)}⊂\mathbb{F}_q^n

For a fixed basis

f1, f2, ..., fk

for L(G) over, the corresponding Goppa code in is spanned over by the vectors

(fi(P1), fi(P2), ..., fi(Pn)).

Therefore

\begin{bmatrix} f_1(P_1) & ... & f_1(P_n) \\ ... & ... & ... \\ f_k(P_1) & ... & f_k(P_n) \end{bmatrix}

is a generator matrix for C(D,G)

Equivalently, it is defined as the image of

,

where f is defined by .

The following shows how the parameters of the code relate to classical parameters of linear systems of divisors D on C (cf. Riemann–Roch theorem for more). The notation l(D) means the dimension of L(D).

Proposition A The dimension of the Goppa code C(D,G) is

,

Proposition B The minimal distance between two code words is

.

Proof A

Since

we must show that

.

Suppose . Then f(P_i)=0, i=1, \dots ,n, so . Thus, f \in L(G-D).
Conversely, suppose .
Then

since

.

(G doesn't “fix” the problems with the, so f must do that instead.) It follows that

.

Proof B
To show that, suppose the Hamming weight of is d. That means that for s, say . Then f \in L(G-P_{i_1} - \dots - P_{i_{n-d}}), and

.

Taking degrees on both sides and noting that

,

we get

,

so

. Q.E.D.

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