Goppa 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" by using a number of fixed rational points

:"P"₁, "P"₂, ..., "P"_n

of X defined over $mathbb\{F\}\_q$, and 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. 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 $P\_i$'s is constructed as follows.

For a fixed basis

:"f"₁, "f"₂, ..., "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"("P"₁), "f"_"i"("P"₂), ..., "f"_"i"("P"_n)).

Equivalently, it is defined as the image of

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

where "f" is defined by $f\; longmapsto\; (f(P\_1),\; dots\; ,f(P\_n))$.

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").

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

:$k\; =\; l(G)\; -\; l(G-D)$,

and the minimal distance between two code words is

:$d\; geq\; n\; -\; deg(G)$.

Proof

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, $f\; inL(G-D)$. Conversely, suppose $f\; in\; L(G-D)$. Then

:$mathrm\{div\}(f)>\; D$

since

:.

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

:$f(P\_i)=0,\; i=1,\; dots\; ,n$.

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

:$mathrm\{div\}(f)+G-P\_\{i\_1\}\; -\; dots\; -\; P\_\{i\_\{n-d>\; 0$.

Taking degrees on both sides and noting that

:$deg(mathrm\{div\}(f))=0$,

we get

:$deg(G)-(n-d)\; geq\; 0$,

so

:$d\; geq\; n\; -\; deg(G)$. Q.E.D.

Residue Code

The residue code can be defined as the dual of the function code, or as the residue of some functions at the $P\_i$'s.

Applications

In cryptography, Goppa codes are used in the McEliece cryptosystem.

In general, Goppa codes are considered 'good' linear codes, in that they permit the correction of

:$\{n^k\}\; choose\; \{log\_2\; n\}$

errors. Also their decoding is easy, using the Euclidean algorithm and Berlekamp-Massey algorithm, in particular.

External links

* [http://upload.wikimedia.org/wikibooks/en/7/71/Algebraic_Geometric_Coding_Theory.pdf An undergraduate thesis on Algebraic Geometric Coding Theory]