Galois-qudit RS code[1]
Description
An \([[n,k,n-k+1]]_q\) (with \(q>n\)) Galois-qudit CSS code constructed using two RS codes over \(GF(q)\).
Let \(C_1\) be a \([n,k_1,d_1]_q\) RS code and \(C_2^\perp\) be a \([n,k_2,d_2]_q\) RS 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 \(GF(q)\setminus \{0\}\).'
Cousins
- Folded quantum RS (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 are CSS codes constructed from RS codes.
- Quantum maximum-distance-separable (MDS) code— A polynomial code is a quantum MDS code when \(n-k_1=k_1-k_2\).
- Concatenated quantum code— Recursive concatenations of quantum RS codes can be asymptotically good [2].
- Asymmetric quantum code— Asymmetric Galois-qudit RS codes have been constructed [3,4].
- Perfect-tensor code— AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [5–7]. MDS RS codes can yield perfect tensors via the CSS and Hermitian constructions [8] (see also Refs. [9,10]).
- Quantum tensor-product code— Product codes constructed from a self-orthogonal and an arbitrary RS code yields an RS code [11].
- Approximate secret-sharing code— Polynomial codes can be used for a specific construction of this code.
Primary Hierarchy
References
- [1]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [2]
- Z. Li, L.-J. Xing, and X.-M. Wang, “A family of asymptotically good quantum codes based on code concatenation”, (2008) arXiv:0901.0042
- [3]
- La Guardia, G. G., R. Palazzo, and C. Lavor. "Nonbinary quantum Reed-Solomon codes." Int. J. Pure Applied Math 65.1 (2010): 55-63.
- [4]
- G. G. La Guardia, “Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes”, Quantum Information Processing 11, 591 (2011) DOI
- [5]
- A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
- [6]
- Z. Raissi, C. Gogolin, A. Riera, and A. Acín, “Optimal quantum error correcting codes from absolutely maximally entangled states”, Journal of Physics A: Mathematical and Theoretical 51, 075301 (2018) arXiv:1701.03359 DOI
- [7]
- D. Alsina and M. Razavi, “Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters”, Physical Review A 103, (2021) arXiv:1907.11253 DOI
- [8]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [9]
- M. Marcolli, “Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces”, (2018) arXiv:1801.09623
- [10]
- M. Heydeman, M. Marcolli, S. Parikh, and I. Saberi, “Nonarchimedean Holographic Entropy from Networks of Perfect Tensors”, (2018) arXiv:1812.04057
- [11]
- M. Grassl and M. Rotteler, “Quantum block and convolutional codes from self-orthogonal product codes”, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005. 1018 (2005) arXiv:quant-ph/0703181 DOI
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