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].
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 \((G\times G) / G\) [2; 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 Cyclic 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 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
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 MDS Gray Evaluation GRS Cyclic 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]
- Diaconis, Persi. “Group representations in probability and statistics.” Lecture notes-monograph series 11 (1988): i-192.
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