Grassmannian code

Grassmannian code[1–3]

Description

Encodes \(K\) states (codewords) into a compact Grassmannian, which includes the real, complex, or quaternionic Grassmannians. Points in a real (complex) Grassmannian index fixed-dimension subspaces of real (complex) vector spaces.

Protection

Optimal Grassmannian codes are robust to coordinate erasure [4,5]. Various code bounds have been formulated [6–10].

Realizations

Multiple description coding [4].Digital fingerprinting [11].

Notes

Tables of real Grassmannian codes [12]

Cousins

\(t\)-design— Designs have been formulated on Grassmannians [13–16].
Barnes-Wall (BW) lattice— BW lattices support Grassmannian 6-designs [15].
Spacetime code (STC)— MIMO channel capacity when the channel is unknown to the sender and receiver [17,18] can be interpreted as a problem of placing points on the Grassmannian [19].
Constant-dimension code— The finite-field Grassmanian is a finite analogue of the compact Grassmannians.

Member of code lists

Primary Hierarchy

Classical Domain

Homogeneous-space Kingdom

Homogeneous-space codeECC

Parents

Homogeneous-space code

Homogeneous spaces \(G/H\) reduce to real Grassmannians for \(G = O(p+q)\) and \(H = O(p)\times O(q)\), to complex Grassmannians for \(G = U(p+q)\) and \(H = U(p)\times U(q)\), and to quaternionic Grassmannians for \(G = Sp(p+q)\) and \(H = Sp(p)\times Sp(q)\).

Grassmannian code

Children

Alternating projection code

Complex projective space code

Complex projective spaces \(\mathbb{C}P^N\) are complex Grassmannians \(G/H\) for \(G = U(N+1)\) and \(H = U(N)\times U(1)\).

Real projective space code

Real projective spaces \(\mathbb{R}P^N\) are real Grassmannians \(G/H\) for \(G = O(N+1)\) and \(H = O(N)\times O(1)\).

References

[1]: A. R. Calderbank, R. H. Hardin, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces”, (2002) arXiv:math/0208002
[2]: P. W. Shor and N. J. A. Sloane, “A Family of Optimal Packings in Grassmannian Manifolds”, (2002) arXiv:math/0208003
[3]: J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing Lines, Planes, etc.: Packings in Grassmannian Space”, (2002) arXiv:math/0208004
[4]: T. Strohmer and R. Heath, “Grassmannian Frames with Applications to Coding and Communication”, (2003) arXiv:math/0301135
[5]: B. G. Bodmann, “Optimal linear transmission by loss-insensitive packet encoding”, Applied and Computational Harmonic Analysis 22, 274 (2007) DOI
[6]: A. Barg and D. Yu. Nogin, “Bounds on packings of spheres in the Grassmann manifold”, IEEE Transactions on Information Theory 48, 2450 (2002) DOI
[7]: O. Henkel, “Sphere-packing bounds in the Grassmann and Stiefel manifolds”, IEEE Transactions on Information Theory 51, 3445 (2005) arXiv:math/0308110 DOI
[8]: C. Bachoc, “Linear programming bounds for codes in grassmannian spaces”, IEEE Transactions on Information Theory 52, 2111 (2006) arXiv:math/0610812 DOI
[9]: C. Bachoc, Y. Ben-Haim, and S. Litsyn, “Bounds for codes in products of spaces, Grassmann and Stiefel manifolds”, (2006) arXiv:math/0610813
[10]: A. Roy, “Bounds for codes and designs in complex subspaces”, (2008) arXiv:0806.2317
[11]: D. G. Mixon, C. J. Quinn, N. Kiyavash, and M. Fickus, “Fingerprinting with Equiangular Tight Frames”, (2011) arXiv:1111.3376
[12]: Sloane, N. J. A. “How to pack lines, planes, 3-spaces, etc.” Online]: http://www2. research. att. com/ njas/grass/index. html (2006).
[13]: C. Bachoc, R. Coulangeon, and G. Nebe, “Designs in Grassmannian Spaces and Lattices”, Journal of Algebraic Combinatorics 16, 5 (2002) DOI
[14]: C. Bachoc, E. Bannai, and R. Coulangeon, “Codes and designs in Grassmannian spaces”, Discrete Mathematics 277, 15 (2004) DOI
[15]: C. Bachoc, “Designs, groups and lattices”, (2007) arXiv:0712.1939
[16]: A. Breger, M. Ehler, M. Gräf, and T. Peter, “Cubatures on Grassmannians: Moments, Dimension Reduction, and Related Topics”, Applied and Numerical Harmonic Analysis 235 (2017) arXiv:1705.02978 DOI
[17]: B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in Rayleigh flat fading”, IEEE Transactions on Information Theory 46, 543 (2000) DOI
[18]: T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multiple-antenna communication link in Rayleigh flat fading”, IEEE Transactions on Information Theory 45, 139 (1999) DOI
[19]: Lizhong Zheng and D. N. C. Tse, “Communication on the Grassmann manifold: a geometric approach to the noncoherent multiple-antenna channel”, IEEE Transactions on Information Theory 48, 359 (2002) DOI

Page edit log

Victor V. Albert (2025-10-27) — most recent

Cite as:

“Grassmannian code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/grassmannian

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/homogeneous/grassmann/grassmannian.yml.