Stabilizer code over \(GF(q^2)\)[1]

An \([[n,k,d]]_q\) Galois-qudit stabilizer code constructed from a classical code over \(GF(q^2)\) using the one-to-one correspondence between the Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\).

An \(n\) Galois-qudit Pauli stabilizer can be represented as a length-\(n\) vector over \(GF(q^2)\). The stabilizer commutation condition corresponds to a zero trace-alternating inner product between the corresponding vectors. Stabilizer codes over \(GF(q^2)\) can thus be constructed from classical trace-alternating self-orthogonal additive codes over \(GF(q^2)\) [1]. Hermitian self-orthogonal linear codes over \(GF(q^2)\) are automatically trace-alternating self-orthogonal, and applying this construction to such codes yields a class of true stabilizer codes.