# 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

## 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.