Binary quantum Goppa code[1–3]

Description

A quantum AG code obtained from algebraic-geometric Goppa codes via the Galois-qudit CSS construction.

For \(q=2^m\), let \(F/\mathbb{F}_q\) be an algebraic function field of one variable with an automorphism \(\sigma\) of order two fixing \(\mathbb{F}_q\), and let \(P_1,\ldots,P_n\) be pairwise distinct degree-one places such that \(\sigma P_i \neq P_j\) for all \(i,j\). If \(\eta\) is a differential with \(v_{P_i}(\eta)=v_{\sigma P_i}(\eta)=-1\), \(\operatorname{res}_{P_i}(\eta)=1\), and \(\operatorname{res}_{\sigma P_i}(\eta)=-1\), and if \(G\) is a \(\sigma\)-invariant divisor satisfying \(v_{P_i}(G)=v_{\sigma P_i}(G)=0\), then \begin{align} C(G)=\{(f(P_1),\ldots,f(P_n),f(\sigma P_1),\ldots,f(\sigma P_n))\,|\,f\in\mathcal{L}(G)\}\subseteq \mathbb{F}_q^{2n} \tag*{(1)}\end{align} obeys \(C(G)^{\perp_s}=C(H)\), where \(H=(P_1+\cdots+P_n+\sigma P_1+\cdots+\sigma P_n)-G+(\eta)\) [2; Ch. 5]. Whenever \(G \geq H\), this yields an \([[n,k,d]]_q\) quantum stabilizer code with \begin{align} k=\dim G-\dim(G-P_1-\cdots-P_n-\sigma P_1-\cdots-\sigma P_n)-n \tag*{(2)}\end{align} and \(d \geq n-\lfloor \deg G/2 \rfloor\) [1,3][2; Ch. 5].