Stiefel code[1]
Description
Encodes \(K\) states (codewords) into a Stiefel manifold. Points in a real (complex) Stiefel manifold index bases for fixed-dimension subspaces of real (complex) vector spaces.Cousins
- Spacetime code (STC)— Stiefel codes can be thought of as spacetime codes [1]
- Approximate quantum error-correcting code (AQECC)— Riemannian optimization techniques can be applied to design approximate QECCs since the set of unitary encoding maps \(U\) forms a Stiefel manifold [4].
Member of code lists
Primary Hierarchy
Parents
Homogeneous spaces \(G/H\) reduce to real Stiefel manifolds for \(G = O(n)\) and \(H = O(n-k)\), to complex Stiefel manifolds for \(G = U(n)\) and \(H = U(n-k)\), and to quaternionic Stiefel manifolds for \(G = Sp(n)\) and \(H = Sp(n-k)\).
Stiefel code
References
- [1]
- M. T. Hussien, K. G. Seddik, R. H. Gohary, M. Shaqfeh, H. Alnuweiri, and H. Yanikomeroglu, “Space-Time Block Codes over the Stiefel Manifold”, 2015 IEEE Global Communications Conference (GLOBECOM) 1 (2015) DOI
- [2]
- O. Henkel, “Sphere-packing bounds in the Grassmann and Stiefel manifolds”, IEEE Transactions on Information Theory 51, 3445 (2005) arXiv:math/0308110 DOI
- [3]
- C. Bachoc, Y. Ben-Haim, and S. Litsyn, “Bounds for codes in products of spaces, Grassmann and Stiefel manifolds”, (2006) arXiv:math/0610813
- [4]
- M. Casanova, K. Ohki, and F. Ticozzi, “Finding Quantum Codes via Riemannian Optimization”, (2024) arXiv:2407.08423
Page edit log
- Victor V. Albert (2025-11-13) — most recent
Cite as:
“Stiefel code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/stiefel
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/homogeneous/stiefel.yml.