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

This paper studies cyclic, quantum, and DNA codes over the mixed-characteristic ring ${\mathcal{R}}_{p,\alpha }={\mathbb{Z}}_{p^2}[u]/⟨u^2-\alpha ⟩,$ where $p$ is an odd prime and $\alpha \in {\mathbb{F}}_p^{\ast }$. 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.