Root code for the Group Kingdom
Description
Encodes \(K\) states (codewords) using symbols drawn from a group \(G\), typically with the group operation inducing a natural notion of translation or symmetry on the alphabet. The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice.Cousins
- Group-based quantum code— Group-based quantum codes are quantum counterparts of group-alphabet codes.
- Symmetric-space code— Group spaces for Lie groups \(G\) are symmetric spaces [1; Table 6.1].
- 24-cell code— The 24-cell code has a quaternion-coordinate realization as the 24 elements of the binary tetrahedral group \(2T\), one of the three exceptional finite subgroups of \(SU(2)\) [2].
- 600-cell code— The 600-cell code has a quaternion-coordinate realization as the 120 elements of the binary icosahedral group \(2I \cong 2.A_5\), one of the three exceptional finite subgroups of \(SU(2)\) [2][3; Ch. 8, pg. 207].
- Disphenoidal 288-cell code— The disphenoidal 288-cell code has a quaternion-coordinate realization as the 48 elements of the binary octahedral group \(2O\), one of the three exceptional finite subgroups of \(SU(2)\) [2; Sec. 8.6].
Member of code lists
Primary Hierarchy
Parents
Homogeneous spaces \(G/H\) for trivial \(H\) reduce to group spaces. A group-\(G\) space can also be thought of as a multiplicity-free homogeneous space \((G\times G) / G\) [4; pg. 60].
Group-alphabet code
Children
Analog code alphabets, such as \(\mathbb{R}^n\) or \(\mathbb{C}^n\), are additive groups.
Dihedral codes are group-alphabet codes for the dihedral group \(G=D_n\).
Linear code over \(G\)Lattice Self-dual additive Linear \(q\)-ary Evaluation MDS GRS GRM QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible Self-dual linear Linear code over \(\mathbb{Z}_q\) Gray
Codes in permutations are group-alphabet codes for the symmetric group \(G=S_n\).
Matrix-based codeArray MDS array STC MSRD MRD Constant-weight Combinatorial design Self-dual additive Linear \(q\)-ary Gray Self-dual linear QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible AG Evaluation OA Perfect Nearly perfect Perfect binary GRM MDS GRS Balanced
Matrix-based code alphabets are additive groups.
Ring codeSelf-dual additive AG OA Perfect Balanced Constant-weight Combinatorial design Nearly perfect Perfect binary Linear code over \(\mathbb{Z}_q\) Linear \(q\)-ary Evaluation MDS GRS Gray GRM QR Projective geometry Quantum-inspired classical block Tanner \(q\)-ary LDPC Divisible Self-dual linear
A ring \(R\) is an Abelian group under addition.
References
- [1]
- M. Berger, A Panoramic View of Riemannian Geometry (Springer Berlin Heidelberg, 2003) DOI
- [2]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [3]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [4]
- P. Diaconis, Group Representations in Probability and Statistics, Lecture Notes-Monograph Series, vol. 11 (Institute of Mathematical Statistics, 1988)
Page edit log
- Victor V. Albert (2022-03-24) — most recent
Cite as:
“Group-alphabet code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/group_classical