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

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.

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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Permanent link:
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.