## Description

Also called a linear stabilizer code. A \([[n,k,d]]_q\) stabilizer code whose stabilizer's symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.

The number of generators \(r\) for a true stabilizer code is a multiple of \(m\) (recall that \(q=p^m\) for Galois qudits). As a result, the number \(k=n-r/m\) of logical qudits is an integer.

Each code can be represented by a stabilizer generator matrix \(H=(A|B)\), where each row \((a|b)\) is the \(GF(q)\)-valued symplectic representation of a stabilizer generator.

A Hermitian self-orthogonal linear \([n,k,d]_{q^2}\) code can be used to construct an \([[n,n-2k]]_q\) true stabilizer code with distance no less than \(d\). This Hermitian construction was first proven via the symplectic representation (showing self-orthogonality under the trace-symplectic inner product; see Ref. [1], Corr. 1), and later proven via the stabilizer-over-\(GF(q^2)\) construction (showing self-orthogonality under the trace-alternating inner product; see Ref. [2], Corr. 19). There is an isomorphism between the symplectic and stabilizer-over-\(GF(q^2)\) representations (Ref. [4], Thm. 27.3.8). The Hermitian construction has been extended to \(q^{2m}\)-ary Hermitian self-orthogonal linear codes [5] and similar constructions exist [6].

## Protection

## Parent

## Children

- \([[5,1,3]]_q\) Galois-qudit code
- Galois-qudit BCH code — Galois-qudit BCH codes constructed via the CSS construction are Galois-qudit CSS codes, and the rest are true stabilizer codes.
- Galois-qudit CSS code — Galois-qudit CSS codes are true stabilizer codes [3].
- Galois-qudit GRS code — Galois-qudit GRS codes constructed via the CSS construction are Galois-qudit CSS codes, and the rest are true stabilizer codes.
- Stabilizer code over \(GF(q^2)\) — Trace-alternating self-orthogonal linear codes over \(GF(q^2)\) are equivalent to a class of true stabilizer codes [3]. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal and can be used to construct true stabilizer codes via the stabilizer-over-\(GF(q^2)\) construction ([2], Corr. 19).

## Cousins

- Linear \(q\)-ary code — A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.
- Dual linear code — Hermitian self-orthogonal linear codes over \(GF(q^2)\) yield true stabilizer codes via either the symplectic representation (showing self-orthogonality under the trace-symplectic inner product; see Ref. [1], Corr. 1) or the stabilizer-over-\(GF(q^2)\) construction (showing self-orthogonality under the trace-alternating inner product; see Ref. [2], Corr. 19 or Ref. [4], Thm. 27.3.8).
- Quantum maximum-distance-separable (MDS) code — Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction [7][8][9][10], in particular from cyclic [11], constacyclic [12][13] and negacyclic [14] codes.
- Matrix-product code — Hermitian self-orthogonal matrix-product codes over \(GF(q^2)\) can be used to construct true stabilizer codes [15][16].

## References

- [1]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [2]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [3]
- D. Gottesman. Surviving as a quantum computer in a classical world
- [4]
- W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [5]
- C. Galindo and F. Hernando, “On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes”, (2021) arXiv:2012.11998
- [6]
- M. F. Ezerman, S. Ling, and P. Sole, “Additive Asymmetric Quantum Codes”, IEEE Transactions on Information Theory 57, 5536 (2011) arXiv:1002.4088 DOI
- [7]
- M. GRASSL, T. BETH, and M. RÖTTELER, “ON OPTIMAL QUANTUM CODES”, International Journal of Quantum Information 02, 55 (2004) arXiv:quant-ph/0312164 DOI
- [8]
- 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
- [9]
- X. He, L. Xu, and H. Chen, “New \(q\)-ary Quantum MDS Codes with Distances Bigger than \(\frac{q}{2}\)”, (2015) arXiv:1507.08355
- [10]
- L. Lu et al., “New Quantum MDS codes constructed from Constacyclic codes”, (2018) arXiv:1803.07927
- [11]
- G. G. La Guardia, “New Quantum MDS Codes”, IEEE Transactions on Information Theory 57, 5551 (2011) DOI
- [12]
- X. Kai, S. Zhu, and P. Li, “Constacyclic Codes and Some New Quantum MDS Codes”, IEEE Transactions on Information Theory 60, 2080 (2014) DOI
- [13]
- B. Chen, S. Ling, and G. Zhang, “Application of Constacyclic Codes to Quantum MDS Codes”, IEEE Transactions on Information Theory 61, 1474 (2015) DOI
- [14]
- X. Kai and S. Zhu, “New Quantum MDS Codes From Negacyclic Codes”, IEEE Transactions on Information Theory 59, 1193 (2013) DOI
- [15]
- M. Cao and J. Cui, “Construction of new quantum codes via Hermitian dual-containing matrix-product codes”, Quantum Information Processing 19, (2020) DOI
- [16]
- X. Liu, H. Liu, and L. Yu, “On New Quantum Codes From Matrix Product Codes”, (2021) arXiv:1604.05823

## Page edit log

- Victor V. Albert (2022-07-22) — most recent
- Daniel Gottesman (2022-02-17)
- Victor V. Albert (2022-02-17)

## Cite as:

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