\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code[1]
Description
A member of the \([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) family of Majorana stabilizer codes for \(m \geq 3\) constructed a self-orthogonal first-order RM code (whose dual is the extended Hamming code). A shortened \([[2^{m-1}-1,2^{m-1}-m-2,3]]_{f}\) version can also be defined [2; Prop. 2.5.1]. The logical subspace of the \([[8,3,4]]_{f}\) Hamming Majorana code is a Cartan subspace of the \(E_8\) Lie algebra [3].Cousins
- \([2^m,2^m-m-1,4]\) Extended Hamming code— A Hamming Majorana code is constructed from a first-order RM code (whose dual is the extended Hamming code).
- \([2^m,m+1,2^{m-1}]\) First-order RM code— A Hamming Majorana code is constructed from a first-order RM code (whose dual is the extended Hamming code).
- \(E_8\) Gosset lattice— The logical subspace of the \([[8,3,4]]_{f}\) Hamming Majorana code is a Cartan subspace of the \(E_8\) Lie algebra [3].
- \([[4,2,2]]\) Four-qubit code— The \([[8,3,4]]_{f}\) Hamming Majorana code is a Majorana stabilizer code obtained by combining two four-qubit codes [3].
- Majorana color code— The \([[8,3,4]]_{f}\) Hamming Majorana code can be placed onto patches of a 2D lattice to form a Majorana color code [4; Table I].
Primary Hierarchy
Parents
A Hamming Majorana code is constructed from a first-order RM code (whose dual is the extended Hamming code).
Small-distance qubit stabilizer codeStabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum
\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code
References
- [1]
- M. B. Hastings, “Small Majorana Fermion Codes”, (2017) arXiv:1703.00612
- [2]
- R. Okada, “A Quantum Analog of Delsarte’s Linear Programming Bounds”, (2025) arXiv:2502.14165
- [3]
- P. Lévay and F. Holweck, “A fermionic code related to the exceptional group E \({}_{\text{8}}\)”, Journal of Physics A: Mathematical and Theoretical 51, 325301 (2018) arXiv:1801.06998 DOI
- [4]
- D. Litinski and F. von Oppen, “Quantum computing with Majorana fermion codes”, Physical Review B 97, (2018) arXiv:1801.08143 DOI
Page edit log
- Victor V. Albert (2025-01-27) — most recent
Cite as:
“\([[2^{m-1},2^{m-1}-m-1,4]]_{f}\) Hamming Majorana code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/majorana_hamming