\([[36,6,6]]\) mirror code[1]
Description
Abelian mirror code on \(G=\mathbb{Z}_6\times\mathbb{Z}_6\) with check weight six that is not equivalent to a CSS code via Hadamard gates [1; Fig. 1]. Its symplectic double is the half-gross code.
The subsets defining the code are \(A=\{(1,2),(4,3),(4,4)\}\) and \(B=\{(2,4),(3,1),(4,1)\}\) [1; Fig. 1]. Qubits form a \(6\times 6\) periodic square lattice, and each stabilizer generator acts as \(Z\) on a translate of \(A\) and as \(X\) on the mirror-image translate of \(B\). Table 1 of Ref. [1] lists an equivalent presentation over \(\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_3\). The code is not equivalent to a CSS code under local Clifford gates and qubit permutations.
Cousin
- \([[72,12,6]]\) half-gross code— The symplectic double of the \([[36,6,6]]\) mirror code is the half-gross code [2]. The other symplectic halves of the half-gross code have parameters \([[36,6,5]]\) and \([[36,6,3]]\).
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References
- [1]
- A. B. Khesin and J. Z. Lu, “Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime”, (2026) arXiv:2603.05496
- [2]
- V. V. Albert, unpublished computation, 2026
Page edit log
- Victor V. Albert (2026-09-26) — most recent
Cite as:
“\([[36,6,6]]\) mirror code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/mirror_36_6_6, arXiv:2606.11484