Binary quantum Goppa code[1][2]

Description

Also known as a quantum AG code. Binary quantum Goppa codes are a family of \( [[n,k,d]]_{GF(q)} \) CSS codes for \( q=2^m \), generated 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 \([[n,k,d]]_{GF(q)}\) quantum stabilizer code such that \(k = \text{dim} G - \text{dim}(G-P_1 - \cdots - P_n - \sigma P_1 - \cdots - \sigma P_n) - n~.\)

Protection

Protects against weight \(t\) errors where \( 0 < t \leq \lfloor \frac{d^*-g-1}{2} \rfloor \) where \( d^* = \text{deg} G + 2 -2g \) and \(g\) is the genus of the function field and \(d \geq n - \lfloor \frac{deg G}{2} \rfloor\).

Encoding

Encoding defined in Ref. [3] uses a technique from Ref. [4] to encode quantum stabilizer codes.

Decoding

Farran algorithm [5].

Parent

Cousin

  • Goppa code — Classical Goppa codes are used to construct their quantum versions.

Zoo code information

Internal code ID: binary_quantum_goppa

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: binary_quantum_goppa

Cite as:
“Binary quantum Goppa code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_quantum_goppa
BibTeX:
@incollection{eczoo_binary_quantum_goppa, title={Binary quantum Goppa code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/binary_quantum_goppa} }
Permanent link:
https://errorcorrectionzoo.org/c/binary_quantum_goppa

References

[1]
Annika Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”. quant-ph/0501074
[2]
A. Niehage, “Nonbinary Quantum Goppa Codes Exceeding the Quantum Gilbert-Varshamov Bound”, Quantum Information Processing 6, 143 (2006). DOI
[3]
R. Matsumoto, “Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes”, IEEE Transactions on Information Theory 48, 2122 (2002). DOI; quant-ph/0107129
[4]
Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
[5]
J. I. Farran, “Decoding Algebraic Geometry codes by a key equation”. math/9910151

Cite as:

“Binary quantum Goppa code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/binary_quantum_goppa

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/binary_quantum_goppa.yml.