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
Homogeneous spaces \(G/H\) for trivial \(H\) reduce to group spaces.
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 Evaluation MDS GRS GRM QR Projective geometry Tanner \(q\)-ary LDPC Divisible Self-dual additive 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 code alphabets are fields, which are groups under addition.
Ring codeAG Editing OA Perfect Balanced Constant-weight Combinatorial design Nearly perfect Perfect binary 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