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.
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.
- 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.
- 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 .
- Binary group-orbit code — Binary group-orbit codes can be mapped into Slepian group-orbit codes via various mappings [8; Ch. 8].
- Spherical design code — 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 \(G\) . 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.
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- Victor V. Albert (2022-11-18) — most recent
“Slepian group-orbit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/slepian_group