Linear code

Description

A code whose set of codewords is closed under addition and multiplication by elements of its alphabet, which can be either a field or a ring. In other words, for any codewords \(x,y\), \(\alpha x+ \beta y\) is also a codeword for any alphabet elements \(\alpha,\beta\). This extra structure yields much information about their properties, making them a large and well-studied subset of codes.

Parent

  • Group-orbit code — Since codewords of a linear code form an abelian group under addition, any codeword \(c\) can be obtained from any other codeword via addition of a codeword. This means that the set of codewords can be thought of as an orbit of a particular codeword under the group; e.g., see [1; Thm. 8.4.2] for the binary case. However, group orbit codes do not have to be linear; see [1; Remark 8.4.3].

Children

Cousins

  • Stabilizer code — Linear (stabilizer) codes form a large and well-studied subset of all classical (quantum) codes because features such as decoding and level of protection are typically easier to determine than those of nonlinear (non-stabilizer) codes.
  • Lattice-based code — Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals.
  • Linear code over \(G\) — Since linear codes over \(G\) for abelian (non-Abelian) groups are closed under addition (multiplication), such codes can be thought of as linear codes over groups.

References

[1]
T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
[2]
J. J. Bernal, Á. del Río, and J. J. Simón, “An intrinsical description of group codes”, Designs, Codes and Cryptography 51, 289 (2009). DOI
[3]
W. C. Huffman, J.-L. Kim, and P. Solé, Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021). DOI
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Zoo code information

Internal code ID: linear

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Zoo Code ID: linear

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

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/classical/properties/linear.yml.