Galois-qudit polynomial code (QPyC)[1]

Description

Also called quantum Reed-Solomon code. An \([[n,k,n-k+1]]_{GF(q)}\) (with \(q>n\)) Galois-qudit CSS code constructed using two Reed-Solomon codes over \(GF(q)\). Let \(C_1\) be a \([n,k_1,d_1]_q\) Reed-Solomon code and \(C_2^\perp\) be a \([n,k_2,d_2]_q\) Reed-Solomon code, modified such that \(C_2^\perp \subseteq C_1\) and \(0\le k_2 \le k_1 \le n\). Then, a polynomial code is a non-degenerate \([[n,k_2,d]]_{GF(q)}\) Galois-qudit CSS code with \(d=\min(n-k_1+1,k_1-k_2+1)\). The polynomial code is the span of the basis codewords over GF(\(q\)) \begin{align} |\overline{\beta_0,\cdots,\beta_{k_2-1}}\rangle = \sum_{(\beta_{k_2},\cdots,\beta_{k_1-1})\in GF(q) } \bigotimes_{i=1}^{n} \left|\sum_{j=0}^{k_1-1} \beta_j \alpha_i^j \right\rangle, \end{align} where \((\alpha_1, \cdots, \alpha_n)\) are \(n\) distinct points chosen for code \(C_1\) from \(GF(q)\setminus \{0\}\).

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Zoo code information

Internal code ID: galois_polynomial

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Zoo Code ID: galois_polynomial

Cite as:
“Galois-qudit polynomial code (QPyC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_polynomial
BibTeX:
@incollection{eczoo_galois_polynomial, title={Galois-qudit polynomial code (QPyC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_polynomial} }
Permanent link:
https://errorcorrectionzoo.org/c/galois_polynomial

References

[1]
Dorit Aharonov and Michael Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”. quant-ph/9906129

Cite as:

“Galois-qudit polynomial code (QPyC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_polynomial

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