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

An \([[n,k,d]]\) stabilizer code constructed from a quaternary classical code using the one-to-one correspondence between the four Pauli matrices \(\{I,X,Y,Z\}\) and the four elements \(\{0,1,\alpha^2,\alpha\}\) of the quaternary field \(GF(4)\).

An \(n\)-qubit Pauli stabilizer can be represented as a length-\(n\) quaternary vector. The stabilizer commutation condition corresponds to a zero trace-Hermitian inner product between the corresponding vectors. Stabilizer codes over \(GF(4)\) can thus be constructed from classical trace-Hermitian self-orthogonal additive quaternary codes and Hermitian self-orthogonal linear quaternary codes (since the latter are automatically trace-Hermitian self-orthogonal). The classical code corresponds to the stabilizer group \(\mathsf{S}\) while its trace-Hermitian dual corresponds to the normalizer \(\mathsf{N(S)}\).

Stabilizer codes over \(GF(4)\) can be constructed as follows. Let \(\phi\) be a bijection from a linear binary subspace to \(GF(4)^n\). Let \(C\) be a trace-Hermitian self-orthogonal additive 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 \(GF(4)\).