Algebraic geometry code

In mathematics, an algebraic geometric code (AG-code), otherwise known as a Goppa code, is a general type of linear code constructed by using an algebraic curve $X$ over a finite field $\mathbb {F} _{q}$ . Such codes were introduced by V. D. Goppa. In particular cases, they can have interesting extremal properties.

Traditionally, an AG-code can be constructed from a non-singular projective curve X over a finite field $\mathbb {F} _{q}$ by using a number of fixed distinct $\mathbb {F} _{q}$ -rational points

${\mathcal {P}}$ := {P₁, P₂, ..., P_n} ⊂ X ( $\mathbb {F} _{q}$ ) on X.

Let G be a divisor on X, with a support that consists of only rational points and that is disjoint from the $P_{i}$ 's. Thus ${\mathcal {P}}$ ∩ supp(G) = Ø
By the Riemann-Roch theorem, there is a unique finite-dimensional vector space, $L(G)$ , with respect to the divisor G. The vector space is a subspace of the function field of X.

There are two main types of AG-codes that can be constructed using the above information.

Function Code

The function code (or dual code) with respect to a curve X, a divisor G and the set ${\mathcal {P}}$ is constructed as follows.
Let $D=P_{1}+P_{2}+\cdots +P_{n}$ , be a divisor, with the $P_{i}$ defined as above. We usually denote a Goppa code by C(D,G). We now know al we need to define the Goppa code:

C(D,G) = {(f(P₁), ..., f(P_n))|f

\in

L(G)}⊂

\mathbb {F} _{q}^{n}

For a fixed basis

f₁, f₂, ..., f_k

for L(G) over $\mathbb {F} _{q}$ , the corresponding Goppa code in $\mathbb {F} _{q}^{n}$ is spanned over $\mathbb {F} _{q}$ by the vectors

(f_i(P₁), f_i(P₂), ..., f_i(P_n)).

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

\alpha :L(G)\longrightarrow \mathbb {F} ^{n}

,

where f is defined by $f\longmapsto (f(P_{1}),\dots ,f(P_{n}))$ .

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

k=l(G)-l(G-D)

,

Proposition B The minimal distance between two code words is

d\geq n-\deg(G)

.

Proof A

Since

C(D,G)\cong L(G)/\ker(\alpha ),

we must show that

\ker(\alpha )=L(G-D)

.

Suppose $f\in \ker(\alpha )$ . Then $f(P_{i})=0,i=1,\dots ,n$ , so $\mathrm {div} (f)>D$ . Thus, $f\in L(G-D)$ .
Conversely, suppose $f\in L(G-D)$ .
Then

\mathrm {div} (f)>D

since

P_{i}<G,i=1,\dots ,n

.

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

f(P_{i})=0,i=1,\dots ,n

.

Proof B
To show that $d\geq n-\deg(G)$ , suppose the Hamming weight of $\alpha (f)$ is d. That means that $f(P_{i})=0$ for $n-d$ $P_{i}$ s, say $P_{i_{1}},\dots ,P_{i_{n-d}}$ . Then $f\in L(G-P_{i_{1}}-\dots -P_{i_{n-d}})$ , and