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Homogeneous-space code

Alternative names: Coset-space code, \(G/H\) code.
Root code for the Homogeneous-space Kingdom

Description

Encodes \(K\) states (codewords) into a homogeneous (a.k.a. coset) space \(G/H\), where \(G\) is a group and \(H\) is a subgroup of \(G\). The space is labeled by cosets of \(H\) in \(G\). Notable groups include compact groups, locally compact Abelian groups, and finite groups.

The two-sphere is a simple example of a homogeneous space with \(G/H = SO(3)/SO(2)\). Any point on the sphere can be obtained from any other point by a proper (i.e., \(SO(3)\)) rotation, which is equivalent to saying that \(SO(3)\) acts transitively. One can then show that any point is invariant under the subgroup of rotations around the axis parallel to the point. This means that one can associate each point with a coset of \(SO(2)\) in \(SO(3)\).

A special class of homogeneous spaces are symmetric spaces, which are homogeneous spaces with an appropriately defined inversion operation. The two-sphere is a symmetric space, and its operation is inversion through the origin. This holds true in higher dimensions, yielding the \(D\)-dimensional spherical symmetric space family \(SO(D+1)/SO(D)\). Simply connected symmetric spaces have been classified by their curvature.

A special class of symmetric spaces are the two-point homogeneous spaces (a.k.a. rank-one symmetric spaces), for which any two points can be mapped to any other two points that are the same distance apart. This is equivalent to saying that \(G\) acts two-transitively.

Protection

The space of normalizable functions on a homogeneous space, \(L^2(G/H)\), carries a representation of \(G\), which can further be decomposed into irreducible representations (irreps) and their multiplicities. Functions on \(G/H\) that transform as an irrep of \(G\) are called harmonics. The decomposition into irreps is multiplicity free for symmetric spaces, meaning that irreps and their internal indices are sufficient to completely label a basis for the space. For example, the integer angular momentum \(J\) and its \(z\)-axis projection \(m\) are sufficient to completely label the spherical harmonics (a.k.a. be a good set of quantum numbers), which in turn can be used to expand any function on the sphere.

Functions that correspond to projections onto irreps are called zonal spherical functions [1]; these correspond to functions on the double coset space \(H \backslash G / H\) [2; Eq. (2.9)] and can be obtained by averaging harmonics over \(G\). The behavior of such functions can be used to obtain bounds on code parameters such as the Levenshtein bound [3–5].

Optimal codes have been formulated for various homogeneous space (see relatives) as well as for quaternionic and octonionic projective spaces [6,7]. Bounds exist for codes on Stiefel manifolds [8,9].

Notes

See Refs. [11,12][10; Ch. 9] for reviews.

Cousins

  • Lattice-based code— The Levenshtein bound [3–5] and Cohn-Elkies LP bound [13] can be derived for sphere packings by thinking of \(\mathbb{R}^n\) as a homogeneous space of the Euclidean group by the orthogonal group, \(E(n)/O(n)\) [14; Ch. XI]. Upper bounds on kissing numbers can be worked out by treating the sphere as a symmetric space [15].
  • Flag-variety code— The flag variety is a finite homogeneous space [16].
  • Homogeneous-space quantum code— Homogeneous-space quantum codes are quantum counterparts of homogeneous-space codes.

Member of code lists

Primary Hierarchy

Classical Domain
Homogeneous-space Kingdom
Parents
Error-correcting code (ECC)
Homogeneous-space code
Children
Group-alphabet codeModulation scheme Polytope Polyhedron Lattice-shell Spherical design Lattice-based Array MDS array STC MRD MSRD Self-dual additive Linear \(q\)-ary Evaluation Self-dual linear QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible AG Editing OA Perfect GRM MDS GRS Balanced Linear code over \(\mathbb{Z}_q\) Gray Constant-weight Combinatorial design Nearly perfect Perfect binary
Homogeneous spaces \(G/H\) for trivial \(H\) reduce to group spaces.
Unitary code
The unitary group is a compact symmetric space \(G/H\) with \(G=U(N)\times U(N)\) and \(H = U(N)\) [12; Table 3].
Two-point homogeneous-space codeSharp configuration Constant-weight Combinatorial design Self-dual additive Linear \(q\)-ary Gray Evaluation Self-dual linear QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible AG Editing OA Perfect Nearly perfect Perfect binary GRM MDS GRS Balanced Polytope Polyhedron Lattice-shell Spherical design
Grassmannian 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 \(H = Sp(p)\times Sp(q)\).
Hyperbolic sphere packing
Hyperbolic space in \(D\) dimensions is a symmetric space \(G/H\) for \(G = SO(D,1)\) the proper Lorentz group and \(H = O(D)\). The hyperbolic plane is the case \(D=2\).
Constant-weight block codeConstant-weight Combinatorial design Balanced
The set of all weight-\(w\) \(q\)-ary strings of length \(n\) forms the nonbinary Johnson space (a.k.a. \(q\)-ary Johnson space), a finite symmetric space \(G/H\) with \(G = S_{q-1} \wr S_n\) [17][18; Sec. 8.8][12; Table 3]. The number of such strings is \({n \choose w} (q-1)^w\). This reduces to the Johnson space for \(q=2\).
Universally optimal codeSharp configuration MDS GRS
Poset code
Ordered Hamming space can be viewed as a finite symmetric space [11,19,20][12; Table 3].
\(q\)-ary code over \(\mathbb{Z}_q\)Constant-weight Combinatorial design Nearly perfect Perfect binary Linear code over \(\mathbb{Z}_q\) Gray
The space of \(q\)-ary codes over \(\mathbb{Z}_q\) under the Lee metric can be viewed as a finite symmetric space \(G/H\) with \(G = D_q \wr S_n\) [21][12; Table 3].

References

[1]
S. Bochner, “Hilbert Distances and Positive Definite Functions”, The Annals of Mathematics 42, 647 (1941) DOI
[2]
F. Scarabotti and F. Tolli, “Harmonic analysis on a finite homogeneous space”, Proceedings of the London Mathematical Society 100, 348 (2009) arXiv:math/0701533 DOI
[3]
V. I. Levenshtein, “On choosing polynomials to obtain bounds in packing problems.” Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978.
[4]
V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
[5]
V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.
[6]
H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
[7]
A. Glazyrin, “Moments of isotropic measures and optimal projective codes”, (2020) arXiv:1904.11159
[8]
O. Henkel, “Sphere-packing bounds in the Grassmann and Stiefel manifolds”, IEEE Transactions on Information Theory 51, 3445 (2005) arXiv:math/0308110 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]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[11]
C. Bachoc, “Semidefinite programming, harmonic analysis and coding theory”, (2010) arXiv:0909.4767
[12]
C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI
[13]
H. Cohn and N. Elkies, “New upper bounds on sphere packings I”, Annals of Mathematics 157, 689 (2003) arXiv:math/0110009 DOI
[14]
Vilenkin, N. I. (1978). Special functions and the theory of group representations (Vol. 22). American Mathematical Soc.
[15]
C. Bachoc and F. Vallentin, “New upper bounds for kissing numbers from semidefinite programming”, Journal of the American Mathematical Society 21, 909 (2007) arXiv:math/0608426 DOI
[16]
V. Lakshmibai and J. Brown, The Grassmannian Variety (Springer New York, 2015) DOI
[17]
H. Tarnanen, M. J. Aaltonen, and J.-M. Goethals, “On the Nonbinary Johnson Scheme”, European Journal of Combinatorics 6, 279 (1985) DOI
[18]
T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, Harmonic Analysis on Finite Groups (Cambridge University Press, 2008) DOI
[19]
W. J. Martin and D. R. Stinson, “Association Schemes for Ordered Orthogonal Arrays and (T, M, S)-Nets”, Canadian Journal of Mathematics 51, 326 (1999) DOI
[20]
A. Barg and P. Purkayastha, “Bounds on ordered codes and orthogonal arrays”, (2009) arXiv:cs/0702033
[21]
J. ASTOLA, “THE LEE-SCHEME AND BOUNDS FOR LEE-CODES”, Cybernetics and Systems 13, 331 (1982) DOI
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Zoo Code ID: homogeneous_space_classical

Cite as:
“Homogeneous-space code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/homogeneous_space_classical
BibTeX:
@incollection{eczoo_homogeneous_space_classical, title={Homogeneous-space code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/homogeneous_space_classical} }
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“Homogeneous-space code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/homogeneous_space_classical

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