Root code for the Group Kingdom
Description
Encodes \(K\) states (codewords) in coordinates labeled by elements of a group \(G\). The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice.Cousin
Member of code lists
Primary Hierarchy
Parents
Group-alphabet code
Children
Analog code alphabets, \(\mathbb{R}^n\) or \(\mathbb{C}^n\), are infinite fields, which are groups under addition.
Linear code over \(G\)Lattice-based Linear \(q\)-ary Gray Evaluation MDS GRS GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible Self-dual additive Self-dual linear \(q\)-ary linear code over \(\mathbb{Z}_q\)
Codes in permutations are group-alphabet codes for the symmetric group \(G=S_n\).
Matrix-based code alphabets are fields, which are groups under addition.
Ring codeAG Editing OA Perfect Nearly perfect Perfect binary Combinatorial design Balanced \(q\)-ary linear code over \(\mathbb{Z}_q\) Linear \(q\)-ary MDS Gray Evaluation GRS GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible Self-dual additive Self-dual linear
A ring \(R\) is an Abelian group under addition.
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