Description
Encodes \(K\) states (codewords) into a symmetric space, which is a homogeneous space \(G/H\) with an additional property whose definition depends on whether the homogeneous space is continuous or Finite.
Continuous symmetric spaces 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 \(SO(D+1)/SO(D)\). Cartan classified the compact symmetric spaces whose \(G\) are simple real Lie groups [1,2]. These spaces include spheres, projective spaces, and Grassmannians. Noncompact symmetric spaces include Euclidean and hyperbolic spaces.
Finite symmetric spaces are defined in coding theory as spaces admitting a generously transitive group action, i.e., for all pairs of points \(x,y \in G/H\), there exists a \(g\in G\) such that \(g.x = y\) and \(g.y = x\) [3; Def. 4.5][4; Sec. 3.4].
Protection
The decomposition of a symmetric space into \(G\)-irreps is multiplicity free. Optimal codes have been formulated for quaternionic and octonionic projective spaces [5,6].Cousins
- Lattice-based code— Upper bounds on kissing numbers can be worked out by treating the sphere as a symmetric space [7].
- Group-alphabet code— Group spaces for Lie groups \(G\) are symmetric spaces [2; Table 6.1].
Member of code lists
Primary Hierarchy
Parents
Continuous symmetric spaces are homogeneous spaces with an appropriately defined inversion operation. Finite symmetric spaces are defined in coding theory as spaces admitting a generously transitive group action [3; Def. 4.5][4; Sec. 3.4]. For multiplicity-free spaces such as symmetric spaces, the zonal spherical functions form an Abelian algebra, and the behavior of such functions can be used to obtain bounds on code parameters such as the Levenshtein bound [8–10].
Symmetric-space code
Children
The permutation group can be viewed as a finite symmetric space \(G/H\) with \(G = S_n \times S_n\) and \(H=S_n\) [11,12][4; Table 3].
The unitary group is a compact symmetric space \(G/H\) with \(G=U(N)\times U(N)\) and \(H = U(N)\) [4; Table 3].
Two-point homogeneous-space codeMRD Sharp configuration Constant-weight Combinatorial design Self-dual additive Linear \(q\)-ary Gray Evaluation Self-dual linear Cyclic QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible AG OA Perfect Nearly perfect Perfect binary GRM MDS GRS Balanced Polytope Polyhedron Lattice-shell Spherical design
A special class of symmetric spaces are the two-point homogeneous spaces (a.k.a. rank-one symmetric spaces [2; Table 6.1]), whose metric is \(G\)-invariant and for which any two points can be mapped, via some \(g\in G\), to any other two points that are the same distance apart [3; Def. 4.7]. This is equivalent to saying that \(G\) acts two-transitively.
Grassmannians are symmetric spaces \(G/H\) for \(G = O(p+q)\) and \(H = O(p)\times O(q)\) in the case of real Grassmannians, \(G = U(p+q)\) and \(H = U(p)\times U(q)\) in the case of complex Grassmannians, and \(G = Sp(p+q)\) and \(H = Sp(p)\times Sp(q)\) in the case of quaternionic Grassmannians.
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 \(G = S_{q-1} \wr S_n\) [13][14; Sec. 8.8][4; Table 3]. The number of such strings is \({n \choose w} (q-1)^w\). This reduces to the Johnson space for \(q=2\).
Ordered Hamming space can be viewed as a finite symmetric space [3,15,16][4; 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 \(G = D_q \wr S_n\) [17,18][4; Table 3].
References
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- E. Cartan, “Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple”, Annales scientifiques de l’École normale supérieure 44, 345 (1927) DOI
- [2]
- M. Berger, A Panoramic View of Riemannian Geometry (Springer Berlin Heidelberg, 2003) DOI
- [3]
- C. Bachoc, “Semidefinite programming, harmonic analysis and coding theory”, (2010) arXiv:0909.4767
- [4]
- 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
- [5]
- H. Cohn, A. Kumar, and G. Minton, “Optimal simplices and codes in projective spaces”, Geometry & Topology 20, 1289 (2016) arXiv:1308.3188 DOI
- [6]
- A. Glazyrin, “Moments of isotropic measures and optimal projective codes”, (2020) arXiv:1904.11159
- [7]
- 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
- [8]
- 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.
- [9]
- V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303
- [10]
- V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.
- [11]
- H. Tarnanen, “Upper Bounds on Permutation Codes via Linear Programming”, European Journal of Combinatorics 20, 101 (1999) DOI
- [12]
- P. J. Dukes, F. Ihringer, and N. Lindzey, “On the Algebraic Combinatorics of Injections and its Applications to Injection Codes”, (2019) arXiv:1912.04500
- [13]
- H. Tarnanen, M. J. Aaltonen, and J.-M. Goethals, “On the Nonbinary Johnson Scheme”, European Journal of Combinatorics 6, 279 (1985) DOI
- [14]
- T. Ceccherini-Silberstein, F. Scarabotti, and F. Tolli, Harmonic Analysis on Finite Groups (Cambridge University Press, 2008) DOI
- [15]
- 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
- [16]
- A. Barg and P. Purkayastha, “Bounds on ordered codes and orthogonal arrays”, (2009) arXiv:cs/0702033
- [17]
- J. ASTOLA, “THE LEE-SCHEME AND BOUNDS FOR LEE-CODES”, Cybernetics and Systems 13, 331 (1982) DOI
- [18]
- P. Sole, “The les association scheme”, Lecture Notes in Computer Science 45 (1988) DOI
Page edit log
- Victor V. Albert (2025-11-14) — most recent
Cite as:
“Symmetric-space code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/symmetric_space