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Matrix-based code

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 \(GF(q)\) or the complex numbers \(\mathbb{C}\)).

Cousin

  • \(q\)-ary code— Elements of fields such as \(GF(p^{ml})\) can be written as \(m\)-dimensional vectors over \(GF(p^l)\) or \((m\times l)\)-dimensional matrices over \(GF(p)\). This idea is used to convert between ordinary block codes and matrix-based codes such as disk array codes and rank-metric codes.

Member of code lists

Primary Hierarchy

Classical Domain
Matrix Kingdom
Parents
Finite-dimensional error-correcting code (ECC)ECC
Block codeECC
Group-alphabet codeECC
Matrix-based code alphabets are fields, which are groups under addition.
Matrix-based code
Children
Distributed computation code
Array codeMDS array
Spacetime code (STC)
Sum-rank-metric codeMRD MSRD
Tensor-product code
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Zoo Code ID: matrices_into_matrices

Cite as:
“Matrix-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/matrices_into_matrices
BibTeX:
@incollection{eczoo_matrices_into_matrices, title={Matrix-based code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/matrices_into_matrices} }
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Permanent link:
https://errorcorrectionzoo.org/c/matrices_into_matrices

Cite as:

“Matrix-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/matrices_into_matrices

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/matrices/matrices_into_matrices.yml.