One point algebraic code
Definition
Let \(S\) be a numerical semigroup, let \(q\) be a power of a prime number, and let \(\mathbb{F}_q\) be the finite field with \(q\) elements. Let \(R\) be the affine coordinate ring of an absolutely irreducible non-singular curve over \(\mathbb{F}_q\) with a single rational point at infinity, say \(Q\). Let \(\mathcal{P} = \{P_1, \ldots, P_n\}\) be a set of \(n\) distinct affine \(\mathbb{F}_q-\)rational points of the curve. It is defined the evaluation map on \(\mathcal{P}\) as
\[ ev_{\mathcal{P}} : R \to \mathbb{F}_q^n ~ ; ~ ev_{\mathcal{P}}(f) = (f(P_1), \ldots, f(P_n)). \]
Set \(\mathcal{L}(mQ) = \{f \in R ~ | ~ v_Q(f) \ge -m\}\), where \(v_Q\) is the discrete valuation at \(Q\). The image of \(-v_Q\) of \(R^*\) is a numerical semigroup, say \(S\). Given \(s \in S\), it is defined the one point algebraic code as the orthogonal (with respect to the usual dot product) of \(ev_{\mathcal{P}}(\mathcal{L}(sQ))\).