Alternative names: Two-dimensional code.
Root code for the Matrix Kingdom
Description
Encodes \(K\) states (codewords) in an \(m\times n\)-dimensional matrix of coordinates over a field (e.g., the Galois field \(\mathbb{F}_q\) or the complex numbers \(\mathbb{C}\)).Cousin
- Two-point homogeneous-space code— Matrices over \(\mathbb{F}_q\) form a finite two-point homogeneous space [1; Table 2].
Member of code lists
Primary Hierarchy
Parents
Matrix-based code alphabets are fields, which are groups under addition.
Matrix-based code
Children
Subspace codes are represented my generator matrices of subspaces of \(\mathbb{F}_q^n\).
\(q\)-ary codeConstant-weight Combinatorial design Self-dual additive Linear \(q\)-ary Gray Evaluation Self-dual linear QR Projective geometry Tanner \(q\)-ary LDPC Divisible AG Editing OA Perfect Nearly perfect Perfect binary GRM MDS GRS Balanced
Matrix-based codes over \(\mathbb{F}_q\) whose codewords are vectors reduce to \(q\)-ary codes. Elements of fields such as \(\mathbb{F}_{p^{ml}}\) can be written as \(m\)-dimensional vectors over \(\mathbb{F}_{p^{l}}\) or \((m\times l)\)-dimensional matrices over \(\mathbb{F}_p\). This idea is used to convert between ordinary block codes and matrix-based codes such as disk array codes and rank-metric codes.
References
- [1]
- C. Bachoc, D. C. Gijswijt, A. Schrijver, and F. Vallentin, “Invariant Semidefinite Programs”, International Series in Operations Research & Management Science 219 (2011) arXiv:1007.2905 DOI
Page edit log
- Victor V. Albert (2022-02-16) — most recent
Cite as:
“Matrix-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/matrices_into_matrices