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.

In this paper, we introduced a new class of infinitesimal evaluation codes over the dual-number extension $R={\mathbb{F}}_{q^2}+u{\mathbb{F}}_{q^2},u^2=0,$ obtained by evaluating polynomials at perturbed points $a_i+u{b}_i$. This evaluation produces a coupled value–derivative structure through the identity $f(a_i+u{b}_i)=f_0(a_i)+u(b_i{f}_0^{\prime }(a_i)+f_1(a_i)),$ which enriches classical evaluation codes with first-order infinitesimal corrections. We established the Hermitian duality theory for these codes and showed that Hermitian orthogonality over $R$ decomposes into a residue-layer condition over ${\mathbb{F}}_{q^2}$ together with a correction equation involving the infinitesimal parameters. This yields explicit criteria for Hermitian self-orthogonality. Using these criteria, we constructed several families of Hermitian self-orthogonal infinitesimal evaluation codes, including multiplier perturbation, locator perturbation, and subgroup–coset constructions. Via the Gray map and the Hermitian construction, these codes produce new families of $q$-ary quantum stabilizer codes.

Let $p$ be a prime and $m$ a positive integer. This paper investigates cyclic and self-dual codes of length $n$ over the local Frobenius non-chain ring $R=\mathrm{G}\mathrm{R}(p^3,m)[v]$, with $v^2=p^2\alpha$, $\alpha \in {\mathbb{F}}_{p^m}^{\ast }$, and $pv=0$. First, we characterize the algebraic structure of cyclic codes of arbitrary length over $R$. When $gcd(n,p)=1$, explicit generator polynomials are determined, and the corresponding dual and self-orthogonal structures are derived. A key result of this study is the proof that self-dual cyclic codes do not exist over $R$. In addition, the enumeration formula for cyclic LCD codes is given by $2^{e_1+\frac{e_2}{2}}.$ Several examples and tables are provided to illustrate the theoretical findings and the derived mass formulas for cyclic self-orthogonal and LCD codes.