# Stabilizer code over \(GF(4)\)[1]

## Description

An \([[n,k,d]]\) stabilizer code whose encoding is based on a weakly self-dual additive quaternary code \((n, 2^{n-k}, d^*)_4\) with respect to the trace inner product where \(d \ge d^*\). The quaternary field \(GF(4)=\mathbf{F}_4\) consists of \(\{0, 1, w, \bar{w}\}\), with \(\bar{w} = w^2 = w + 1\), \(\mathrm{Tr}(x) = x+\bar{x}\), and trace inner product \(u * v = \mathrm{Tr}(u \cdot \bar{v})\). There is a mapping \(L\) between Pauli matrices \(I, Y, Z, X\) and \(0, 1, \bar{w}, w\), in turn \([A, B] \Leftrightarrow Tr\langle L(A), L(A)\rangle\). The classical self-dual code \(C\) over \(GF(4)^n\) corresponds to the stabilizer group \(\mathsf{S}\) while \(C^{\perp}\) corresponds to \(\mathsf{N(S)}\).

The quaternary code needs to only be additive (its codewords are closed under addition). It need not be a linear code, which would require the set of codewords to be closed under multiplication as well.

## Protection

## Notes

## Parent

## Child

- \([[5,1,3]]\) perfect code — The \([[5,1,3]]\) code is derived from the \([5,3,3]_4\) Hamming code.

## Cousins

- Additive \(q\)-ary code — 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 \(GF(4)\).
- Stabilizer code over \(GF(q^2)\) — Stabilizer codes over \(GF(q^2)\) are Galois-qudit extensions of those over \(GF(4)\).

## Zoo code information

## References

- [1]
- A. R. Calderbank et al., “Quantum Error Correction via Codes over GF(4)”. quant-ph/9608006

## Cite as:

“Stabilizer code over \(GF(4)\)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/stabilizer_over_gf4