Hermitian Galois-qudit code

Hermitian Galois-qudit code[2–4][1; Corr. 5]

Alternative names: \(GF(q^2)\)-linear stabilizer code.

Description

An \([[n,k,d]]_q\) true Galois-qudit stabilizer code constructed from a Hermitian self-orthogonal linear code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\).

Galois-qudit stabilizer codes are in one-to-one correspondence with trace-alternating self-orthogonal additive codes of length \(n\) over \(GF(q^2)\) via the \(GF(q^2)\) representation. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal, and applying this mapping to such codes yields Hermitian codes [4; Corr. 19].

Protection

A Hermitian self-orthogonal linear \([n,k,d]_{q^2}\) code yields an \([[n,n-2k]]_q\) true stabilizer code with distance no less than \(d\); this is called the Hermitian construction. The Hermitian construction was first proven via the Galois symplectic representation (showing self-orthogonality under the trace-symplectic inner product; see Ref. [3], Corr. 1). There is an isomorphism between the Galois-symplectic and \(GF(q^2)\) representations [5; Thm. 27.3.8].

It has also been extended to \(q^{2m}\)-ary Hermitian self-orthogonal linear codes [6], and similar constructions were formulated in Ref. [7]. Quantum construction X, related to (classical) Construction X and Construction XX, allows for the use of nearly self-orthogonal codes [8–10]; see Ref. [11] for a review.

Cousins

Dual linear code— Hermitian codes are constructed from Hermitian self-orthogonal linear codes over \(GF(q^2)\) via the \(GF(q^2)\) representation.
Matrix-product code— Hermitian self-orthogonal matrix-product codes over \(GF(q^2)\) can be used to construct quantum codes via the Hermitian construction [12,13].
Subsystem Galois-qudit stabilizer code— The Hermitian construction has been extended to subsystem Galois-qudit stabilizer codes [14].
Complex Hadamard spherical code— Complex Hadamard matrices can be used to build Hermitian [15] and other [16] Galois-qudit stabilizer codes.
Quantum maximum-distance-separable (MDS) code— Many quantum MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [17–20], in particular from cyclic [21], constacyclic [20,22,23] and negacyclic [24] codes.
Galois-qudit BCH code— Galois-qudit BCH codes can be constructed via the CSS construction or the Hermitian construction.
Quantum duadic code— Quantum duadic codes can be constructed via the CSS construction or the Hermitian construction.
Galois-qudit quantum RM code— Galois-qudit RM codes can be constructed via the CSS construction or the Hermitian construction.
Galois-qudit GRS code— Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.

Member of code lists

Primary Hierarchy

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

True Galois-qudit stabilizer code

Parents

True Galois-qudit stabilizer code

Hermitian codes are true stabilizer codes because they are based on Hermitian self-orthogonal linear (as opposed to additive) codes over \(GF(q^2)\).

Hermitian Galois-qudit code

Children

Hermitian qubit code

\([[9,1,5]]_3\) quantum Glynn code

Quantum twisted code

References

[1]: E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[2]: J. Bierbrauer and Y. Edel, “Quantum twisted codes”, Journal of Combinatorial Designs 8, 174 (2000) DOI
[3]: A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[4]: A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[5]: M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[6]: C. Galindo and F. Hernando, “On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes”, Designs, Codes and Cryptography 90, 1103 (2022) arXiv:2012.11998 DOI
[7]: M. F. Ezerman, S. Ling, and P. Sole, “Additive Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 57, 5536 (2011) arXiv:1002.4088 DOI
[8]: P. Lisoněk and V. Singh, “Quantum codes from nearly self-orthogonal quaternary linear codes”, Designs, Codes and Cryptography 73, 417 (2014) DOI
[9]: Lisonek, Petr, and Reza Dastbasteh. "Constructions of quantum codes." In 3rd International Workshop on Boolean Functions and their Applications, loen, norway. https://org. uib. no/selmer/workshops/BFA2018/Slides/Lisonek. pdf. 2018.
[10]: M. Frederic Ezerman, M. Grassl, S. Ling, F. Özbudak, and B. Özkaya, “Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes”, IEEE Transactions on Information Theory 71, 499 (2025) arXiv:2405.15057 DOI
[11]: M. Grassl, “Algebraic quantum codes: linking quantum mechanics and discrete mathematics”, International Journal of Computer Mathematics: Computer Systems Theory 6, 243 (2020) arXiv:2011.06996 DOI
[12]: M. Cao and J. Cui, “Construction of new quantum codes via Hermitian dual-containing matrix-product codes”, Quantum Information Processing 19, (2020) DOI
[13]: X. Liu, H. Liu, and L. Yu, “On New Quantum Codes From Matrix Product Codes”, (2021) arXiv:1604.05823
[14]: S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
[15]: R. Egan, “A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes”, (2023) arXiv:2309.07522
[16]: B. Sarac and D. Acar, “A Construction of Quantum Stabilizer Codes from Classical Codes and Butson Hadamard Matrices”, (2024) arXiv:2407.13527
[17]: M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
[18]: R. Li and Z. Xu, “Construction of[[n,n−4,3]]qquantum codes for odd prime powerq”, Physical Review A 82, (2010) arXiv:0906.2509 DOI
[19]: X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
[20]: L. Lu, W. Ma, R. Li, Y. Ma, and L. Guo, “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
[21]: G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
[22]: X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
[23]: B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
[24]: X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI

Page edit log

Victor V. Albert (2022-07-22) — most recent
Victor V. Albert (2022-03-21)

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml.