# 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.

## Child

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

## Cousins

- Stabilizer code over \(GF(4)\) — Let \(\phi\) be a bijection from a linear binary subspace to \(GF(4)^n\). Let \(C\) be an additive self-orthogonal subcode over \(GF(4)\), containing \(2^{n-k}\) vectors, such that there are no vectors of weight less than \(d\) in \(C^{\perp}\setminus C\). Then, any eigenspace of the inverse map \(\phi^{-1}(C)\) is an \([[n, k, d]]\) stabilizer code over \(GF(4)\).
- 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.
- Stabilizer code over \(GF(q^2)\) — Self-orthogonal additive \(q\)-ary codes are used in this construction.

## Zoo code information

## Cite as:

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