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 \(GF(q)\setminus \{0\}\).
Parent
Cousins
- Prime-qudit polynomial code (QPyC) — Polynomial codes can be defined for modular qudits of prime dimension or, more generally, for Galois qudits.
- Reed-Solomon (RS) code — Polynomial codes are CSS codes constructed from Reed-Solomon codes.
- Cyclic code
- Quantum maximum-distance-separable (MDS) code — A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
- Approximate secret-sharing code — Polynomial codes can be used for a specific construction of this code.
Zoo code information
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