## 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 [3; 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. [2], Corr. 1). There is an isomorphism between the Galois-symplectic and \(GF(q^2)\) representations [4; Thm. 27.3.8].

It has also been extended to \(q^{2m}\)-ary Hermitian self-orthogonal linear codes [5], and similar constructions were formulated in Ref. [6]. Construction X and XX have also been adapted to yield quantum codes [7,8].

## Parent

- 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 \(GF(q^2)\).

## Children

## 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 [9,10].
- Subsystem Galois-qudit stabilizer code — The Hermitian construction has been extended to subsystem Galois-qudit stabilizer codes [11].
- Complex Hadamard spherical code — Complex Hadamard matrices can be used to build Hermitian [12] and other [13] Galois-qudit stabilizer codes.
- Quantum maximum-distance-separable (MDS) code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [14–17], in particular from cyclic [18], constacyclic [17,19,20] and negacyclic [21] 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.

## References

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- [2]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [3]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [4]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
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- 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
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- M. F. Ezerman et al., “Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes”, (2024) arXiv:2405.15057
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- M. Cao and J. Cui, “Construction of new quantum codes via Hermitian dual-containing matrix-product codes”, Quantum Information Processing 19, (2020) DOI
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- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
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- 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
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- X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
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- L. Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
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- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
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- X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
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- B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
- [21]
- 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