Quantum Goppa code[1–3]

Description

A Galois-qudit CSS code constructed using two Goppa codes.

Binary quantum Goppa codes are a family of \( [[n,k,d]]_q \) CSS codes for \( q=2^m \), constructed using classical Goppa codes. Let \( F/GF(q) \) be an algebraic function field of one variable, \(\sigma \) be an automorphism of order 2 of \(F\) which leaves \(GF(q)\) invariant, and \(P_1, \cdots, P_n \) be pairwise distinct places of degree one such that \( \sigma P_i = P_j\) for all \(i,j = 1, \cdots, n\). A place \( P_i \) of \( F/ GF(q) \) is the unique maximal ideal of a discrete valuation ring of the function field. Let \( \nu \) be a differential that satisfies the properties guaranteed by the strong approximation theorem of discrete evaluations. Let \(G\) be a divisor such that \( \sigma G = G \) and the discrete valuation corresponding to place \(P_i \) of \(G\) and the discrete valuation corresponding to place \(\sigma P_i \) of \(G\) are both 0 for all \(i\). Then we can define a code \( C(G) = \{ f(P_1), \cdots, f(P_n), f(\sigma P_1), \cdots , f(\sigma P_n) | f \in \mathcal{L}(G)\} \subset GF(q^{2n})\) and a code \(C(H)\) where \(H = (P_1 + \cdots + P_n + \sigma P_1 + \cdots + \sigma P_n) - G + \nu \). Then the dual of \(C(G)\) with respect to the weighted symplectic inner product with weights \(s_i\) on \( GF(q^n) \) is equivalent to \(C(H)\). Therefore, the orthogonal code of \(C(G)\) is generated by \(H\). Using these properties and the assumption that \(H\) is a subgroup of \(G\), we can construct a classical Goppa code \(C(D,G)\), where \(D\) is the sum of all \(P_i\). Using \(C(D,G)\), we can construct a \([[n,k,d]]_q\) quantum stabilizer code such that \(k = \text{dim} G - \text{dim}(G-P_1 - \cdots - P_n - \sigma P_1 - \cdots - \sigma P_n) - n~.\)