# 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

- Divisible code
- Dual linear code
- Evaluation code
- Group-algebra code — A linear code is a group-algebra code for a group \(G\) if and only if \(G\) is isomorphic to a regular subgroup of the code's permutation automorphism group [2][3; Thm. 16.4.7].
- Linear \(q\)-ary code
- Linear binary code
- Low-density generator-matrix (LDGM) code
- Low-density parity-check (LDPC) code
- \(R\)-linear code

## 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

## Page edit log

- Victor V. Albert (2022-07-17) — most recent

## Zoo code information

## 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.