Quantum stabiliser codes via concatenation of idempotents in commutative group algebras

Group algebras offer a rich algebraic perspective for the study of classical codes. Through this lens, idempotents have been shown to characterise properties of their associated classical codes. However, attempts to extend the use of idempotents to the construction of quantum stabiliser codes have remained limited, arguably due to insufficient variation in their coefficients. Therefore, this paper proposes a new method of concatenating idempotents from commutative quaternary group algebras and studies the constructed stabiliser codes. The resulting stabilisers are binary subspaces of the respective group algebras, each generated by a single element whose components are distinct idempotents that together exhibit sufficient coefficient variation. Notably, these generators may be units, which leads to an interesting deviation from the conventional setting of idempotents as zero-divisors. The potential of these generators in stabiliser formalism are studied via an underlying group involution. It is proven that the parameters of the quantum stabiliser codes can be characterised by the idempotents involved.