Quantum and DNA codes from cyclic codes over the ring $ \mathbb{Z}_{p^{2}}[u]/\langle u^{2}-\alpha\rangle $

This paper studies cyclic, quantum, and DNA codes over the mixed-characteristic ring $ \mathcal{R}_{p, \alpha} = \mathbb{Z}_{p^{2}}[u]/\langle u^{2}-\alpha\rangle, $ where $ p $ is an odd prime and $ \alpha\in\mathbb{F}_{p}^{*} $. When $ \alpha $ is a quadratic residue modulo $ p $, the polynomial $ u^{2}-\alpha $ splits over $ \mathbb{Z}_{p^{2}} $ and $ \mathcal{R}_{p, \alpha} $ is a semi-local ring isomorphic to $ \mathbb{Z}_{p^{2}}\oplus\mathbb{Z}_{p^{2}} $. In this decomposable case, every $ \mathcal{R}_{p, \alpha} $-linear cyclic code admits a canonical idempotent decomposition into two cyclic codes over $ \mathbb{Z}_{p^{2}} $, leading to explicit descriptions of generator polynomials, dual codes, and Lee distances. Both the coprime-length case $ \gcd(n, p) = 1 $ and the repeated-root case $ n = p^{s} $ are analyzed, reflecting their distinct ideal-theoretic behavior. An $ \mathbb{F}_{p} $-linear Gray map is constructed that induces a Lee-to-Hamming isometry from $ \mathcal{R}_{p, \alpha}^{n} $ to $ \mathbb{F}_{p}^{4n} $. Using a compatible bilinear form, we show that the Gray image of a Euclidean self-orthogonal cyclic code remains symplectic self-orthogonal over $ \mathbb{F}_{p} $, which enables the construction of $ p $-ary quantum stabilizer codes via the Calderbank–Shor–Steane method. Explicit computations for small parameters illustrate the resulting quantum code parameters and show that several examples meet or improve known bounds. For $ p = 5 $, the Gray map also admits an interpretation suitable for DNA coding. By mapping Gray images to the IUPAC nucleotide alphabet and exploiting the ring involution $ u\mapsto -u $, we obtain reversible DNA codes through blockwise reversal symmetry. Using coterm polynomials, families of reversible DNA codes with prescribed minimum distance and controlled GC-content are constructed. These results demonstrate how cyclic codes over the mixed-characteristic ring $ \mathcal{R}_{p, \alpha} $ can be used to derive quantum and DNA codes through the Gray map and related algebraic structures.