Galois-qudit RS code[1]
Description
Also called a polynomial code (QPyC). An \([[n,k,n-k+1]]_q\) (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 \([[n,k_2,d]]_q\) 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, \tag*{(1)}\end{align} where \((\alpha_1, \cdots, \alpha_n)\) are \(n\) distinct points chosen for code \(C_1\) from \(GF(q)\setminus \{0\}\).'
Parent
Child
- Quantum Reed-Solomon code — Galois-qudit RS codes for prime-dimensional qudits are quantum RS codes.
Cousins
- Folded quantum Reed-Solomon (FQRS) code — A FQRS code with no extra grouping (\(m=1\)) reduces to a Galois-qudit RS code that is CSS.
- Reed-Solomon (RS) code — Galois-qudit RS codes codes are CSS codes constructed from Reed-Solomon codes.
- 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.
References
- [1]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
Page edit log
- Victor V. Albert (2022-01-10) — most recent
- Qingfeng (Kee) Wang (2021-12-20)
Cite as:
“Galois-qudit RS code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_polynomial