Algebraic geometry code

Algebraic geometry codes, often abbreviated AG codes, are a type of linear code that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa constructed these codes for the first time in 1982.

Algebraic geometry codes, often abbreviated AG codes, are a type of linear code that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa constructed these codes for the first time in 1982. ==Construction== In this section the construction of algebraic geometry codes is described. The section starts with the ideas behind Reed–Solomon codes, which are used to motivate the construction of algebraic geometry codes. Formally, Reed–Solomon codes are defined in the following way. Let \mathbb{F}_q=\{\alpha_1, \dots, \alpha_q\} . Set positive integers k \leq n \leq q . Let \mathbb{F}_{q}[x]_{ The Reed–Solomon code RS(q,n,k) is the evaluation code RS(q,n,k) = \left\{ \left( f(\alpha_1), f(\alpha_2), \dots, f(\alpha_n) \right) : f \in \mathbb{F}_{q}[x]_{ Formally, algebraic geometry codes are defined in the following way. Let F / \mathbb{F}_q be an algebraic function field, D = P_1 + \dots + P_n be the sum of n distinct places of F / \mathbb{F}_q of degree one, and G be a divisor with disjoint support from D . The algebraic geometry code C_{\mathcal{L}}(D,G) associated with divisors D and G is defined as C_{\mathcal{L}}(D,G) := \lbrace (f(P_1),\dots,f(P_n)) : f \in \mathcal{L}(G) \rbrace \subseteq \mathbb{F}_q^n. More information on these codes may be found in both introductory texts as well as advanced texts on coding theory. RS(q,n,k) = \mathcal{C}_\mathcal{L}(D,(k-1)P_\infty) where P_\infty is the point at infinity on the projective line \mathbb{P}^1_{\mathbb{F}_q} and D = P_1 + \dots + P_q is the sum of the other \mathbb{F}_q -rational points. The Hermitian curve is given by the equation x^{q+1} = y^q + y considered over the field \mathbb{F}_{q^2} . This curve is of particular importance because it meets the Hasse–Weil bound with equality, and thus has the maximal number of affine points over \mathbb{F}_{q^2} . With respect to algebraic geometry codes, this means that Hermitian codes are long relative to the alphabet they are defined over. The Riemann–Roch space of the Hermitian function field is given in the following statement. For the Hermitian function field \mathbb{F}_{q^2}(x,y) given by x^{q+1} = y^q + y and for m \in \mathbb{Z}^+ , the Riemann–Roch space \mathcal{L}(mP_\infty) is \mathcal{L}(mP_\infty) = \left\langle x^a y^b : 0 \leq b \leq q-1, aq + b(q+1) \leq m \right\rangle , where P_\infty is the point at infinity on \mathcal{H}_q(\mathbb{F}_{q^2}) . With that, the one-point Hermitian code can be defined in the following way. Let \mathcal{H}_q be the Hermitian curve defined over \mathbb{F}_{q^2} . Let P_\infty be the point at infinity on \mathcal{H}_q(\mathbb{F}_{q^2}) , and D = P_1 + \cdots + P_n be a divisor supported by the n := q^3 distinct \mathbb{F}_{q^2} -rational points on \mathcal{H}_q other than P_\infty . The one-point Hermitian code C(D,mP_\infty) is C(D,mP_\infty) := \left\lbrace (f(P_1),\dots,f(P_n)) : f \in \mathcal{L}(mP_\infty) \right\rbrace \subseteq \mathbb{F}_{q^2}^n.

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