# Additive \(q\)-ary code

## Description

A \(q\)-ary code whose codewords are closed under addition, i.e., for any codewords \(x,y\), \(x+y\) is also a codeword.

## Parents

- Galois-field \(q\)-ary code
- Linear code over \(G\) — Additive \(q\)-ary codes are linear over \(G=GF(q)\) since Galois fields are abelian groups under addition.

## Children

- Dual additive code
- Linear \(q\)-ary code — For \(q>2\), additive codes need not be linear since linearity also requires closure under multiplication.

## Cousin

- Galois-qudit stabilizer code — A Galois-qudit stabilizer code is the closest quantum analogue of an additive code over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.

## Page edit log

- Victor V. Albert (2022-05-19) — most recent
- Shuubham Ojha (2022-05-18)

## Cite as:

“Additive \(q\)-ary code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/q-ary_additive