Galois-qudit RS code[1]
Alternative names: Galois-qudit polynomial code (QPyC).
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 \(GF(q)\setminus \{0\}\).'
Protection
Galois-qudit RS codes can be adapted for insertion and deletion noise [2].Transversal Gates
There exists an order \([[n,\Theta(n),\Theta(n)]]_{n^2}\) punctured RS code family that admits transversal \(CCZ\) gates for any three logical qubits [3]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of several qubits; see [4,6–8][5; Sec. 5.3]. This yields a qubit code family that is asymptotically good up to poly-logarithmic factors [3].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 [9].
- Asymmetric quantum code— Asymmetric Galois-qudit RS codes have been constructed [10–12].
- Perfect-tensor code— AME states for even \(n\) are examples of quantum MDS codes with no logical qubits [13–15]. MDS RS codes can yield perfect tensors via the CSS and Hermitian constructions [16] (see also Refs. [17,18]).
- Quantum tensor-product code— Product codes constructed from a self-orthogonal and an arbitrary RS code yields an RS code [19].
- Approximate secret-sharing code— Polynomial codes can be used for a specific construction of this code.
Primary Hierarchy
Parents
Galois-qudit RS codes are constructed via the CSS construction from RS codes, which are evaluation AG codes.
Galois-qudit RS code
Children
Galois-qudit RS codes for prime-dimensional qudits are prime-qudit RS codes.
References
- [1]
- D. Aharonov and M. Ben-Or, “Fault-Tolerant Quantum Computation With Constant Error Rate”, (1999) arXiv:quant-ph/9906129
- [2]
- M. Hagiwara, “Quantum Deletion Codes Derived From Quantum Reed-Solomon Codes”, (2023) arXiv:2306.13399
- [3]
- Z. He, V. Vaikuntanathan, A. Wills, and R. Y. Zhang, “Quantum Codes with Addressable and Transversal Non-Clifford Gates”, (2025) arXiv:2502.01864
- [4]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [5]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [6]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [7]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [8]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [9]
- Z. Li, L.-J. Xing, and X.-M. Wang, “A family of asymptotically good quantum codes based on code concatenation”, (2008) arXiv:0901.0042
- [10]
- La Guardia, G. G., R. Palazzo, and C. Lavor. "Nonbinary quantum Reed-Solomon codes." Int. J. Pure Applied Math 65.1 (2010): 55-63.
- [11]
- S. A. Aly, “On Quantum and Classical Error Control Codes: Constructions and Applications”, (2008) arXiv:0812.5104
- [12]
- G. G. La Guardia, “Asymmetric quantum Reed-Solomon and generalized Reed-Solomon codes”, Quantum Information Processing 11, 591 (2011) DOI
- [13]
- 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
- [14]
- 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
- [15]
- 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
- [16]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [17]
- M. Marcolli, “Holographic Codes on Bruhat--Tits buildings and Drinfeld Symmetric Spaces”, (2018) arXiv:1801.09623
- [18]
- M. Heydeman, M. Marcolli, S. Parikh, and I. Saberi, “Nonarchimedean Holographic Entropy from Networks of Perfect Tensors”, (2018) arXiv:1812.04057
- [19]
- 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