We investigate m-adic residue codes of prime length p over the mixed-alphabet ring \(\mathbb {F}_q S,\) where \(S=\mathbb {F}_q+v\mathbb {F}_q\) with \(v^2=v,\) assuming q is an odd prime power with \(\gcd (p,q)=1\) and \(m\mid (p-1).\) Using the structural decomposition \(\mathbb {F}_q S \cong \mathbb {F}_q^3,\) we describe cyclic codes as principal ideals generated by lifted orthogonal idempotents and classify them into even-like and odd-like families of Class I and Class II, with explicit generators, orthogonality relations, and size formulas. A Gray map from \(\mathbb {F}_q S\) to \(\mathbb {F}_q^3\) is introduced, together with a Lee weight and a Gray-induced bilinear form, yielding Lee distance bounds in terms of the three cyclic constituents over \(\mathbb {F}_q.\) At the idempotent level, we determine dual codes by analyzing the action of the reciprocal automorphism (and, when \(q=r^2,\) the Frobenius automorphism), and derive a combinatorial disjoint-support criterion on index sets that guarantees the required dual containment for both Euclidean and Hermitian inner products. This leads to a unified framework for constructing Euclidean and Hermitian CSS-type quantum stabilizer codes over \(\mathbb {F}_q,\) with parameters expressed in terms of the degrees of the underlying generator polynomials. Explicit examples illustrate the resulting classical and quantum code families.
