Slepian group-orbit code[1–3] 

Description

Spherical code in \(n\) dimensions whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the orthogonal group \(O(n)\) of rotations in \(n\) dimensions, i.e., the automorphism group of spherical codes under the Euclidean distance. Neither the vector nor the group are unique for a given code.

Protection

Code properties depend on the relationship between the group and the initial vector, and the number of codewords is the number of cosets of an initial vector's symmetry subgroup in \(G\) per the orbit-stabilizer theorem. See Refs. [4–7] for allowed code parameters.

Notes

See [9][8; Ch. 8] for more details and code tables.

Parents

Children

  • Permutation spherical code — Permutations and sign changes can be implemented on vectors by orthogonal matrices, so permutation spherical codes are Slepian group-orbit codes.
  • Real-Clifford subgroup-orbit code
  • Torus-layer spherical code (TLSC) — Polyphase codewords can be implemented by acting on the all-ones initial vector by diagonal orthogonal matrices whose entries are the codeword components [8; Ch. 8]. TLSC codes are generalizations of polyphase codes to other initial vectors and are examples of Abelian Slepian-group codes.
  • Polytope code — Vertices of polytope codes typically form an orbit of the polytope's symmetry group.

Cousins

  • Linear binary code — Any length-\(n\) binary linear code can be used to define a diagonal subgroup of \(n\)-dimensional rotation matrices with \(\pm 1\) on the diagonals via the antipodal mapping \(0\to+1\) and \(1\to-1\). The orbit of this subgroup yields the corresponding Slepian group-orbit code; see [8; Thm. 8.5.2].
  • Binary antipodal code — Any length-\(n\) binary linear code can be used to define a diagonal subgroup of \(n\)-dimensional rotation matrices with \(\pm 1\) on the diagonals via the antipodal mapping \(0\to+1\) and \(1\to-1\). The orbit of this subgroup yields the corresponding Slepian group-orbit code; see [8; Thm. 8.5.2].
  • Linear code over \(G\) — Any finite-group code can be mapped to a Slepian group-orbit code by representing the group using orthogonal matrices [2].
  • Binary group-orbit code — Binary group-orbit codes can be mapped into Slepian group-orbit codes via various mappings [8; Ch. 8].
  • Spherical design — Slepian group-orbit codes can form spherical designs [10,11]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the design strength of orbits of \(G\) [12]. Let \(t+1\) be the degree of the lowest-degree \(G\)-invariant polynomial that is not a polynomial in the norm \(\left\Vert x\right\Vert^2\). Then, any orbit under \(G\) forms a Slepian group-orbit code that is also a spherical \(t\)-design.

References

[1]
D. Slepian, “Group Codes for the Gaussian Channel”, Bell System Technical Journal 47, 575 (1968) DOI
[2]
H.-A. Loeliger, “Signal sets matched to groups”, IEEE Transactions on Information Theory 37, 1675 (1991) DOI
[3]
T. Mittelholzer and J. Lahtonen, “Group codes generated by finite reflection groups”, IEEE Transactions on Information Theory 42, 519 (1996) DOI
[4]
Biggs, Norman. Finite groups of automorphisms: course given at the University of Southampton, October-December 1969. Vol. 6. CUP Archive, 1971.
[5]
C. Downey and J. Karlof, “On the existence of<tex>[M, n]</tex>group codes for the Gaussian channel with<tex>[M, n]</tex>odd”, IEEE Transactions on Information Theory 23, 500 (1977) DOI
[6]
J. Karlof and C. Downey, “Odd Group Codes for The Gaussian Channel”, SIAM Journal on Applied Mathematics 34, 715 (1978) DOI
[7]
R. M. Siqueira and S. I. R. Costa, “Flat tori, lattices and bounds for commutative group codes”, Designs, Codes and Cryptography 49, 307 (2008) DOI
[8]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[9]
I. Ingemarsson, “Group Codes for the Gaussian Channel”, Lecture Notes in Control and Information Sciences 73 DOI
[10]
E. BANNAI, “Spherical t-designs which are orbits of finite groups”, Journal of the Mathematical Society of Japan 36, (1984) DOI
[11]
P. de la Harpe and C. Pache, “Spherical designs and finite group representations (some results of E. Bannai)”, European Journal of Combinatorics 25, 213 (2004) DOI
[12]
S. L. Sobolev, “Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations”, Selected Works of S.L. Sobolev 461 DOI
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Zoo Code ID: slepian_group

Cite as:
“Slepian group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/slepian_group
BibTeX:
@incollection{eczoo_slepian_group, title={Slepian group-orbit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/slepian_group} }
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“Slepian group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/slepian_group

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/group_orbit/slepian_group.yml.