@article{Saif2026, 
author = {Sami H. Saif and Shayea Aldossari},
title = {Quantum and DNA codes from cyclic codes over the ring              Z                      p                  2                      [  u  ]      /    ⟨      u          2        −  α  ⟩},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {7497-7528},
keywords = {cyclic codes, Gray map, quantum codes, DNA codes},
url = {https://www.sciopen.com/article/10.3934/math.2026307},
doi = {10.3934/math.2026307},
abstract = {This paper studies cyclic, quantum, and DNA codes over the mixed-characteristic ring              R              p      ,      α        =            Z                      p                  2                      [  u  ]      /    ⟨      u          2        −  α  ⟩  , where    p is an odd prime and    α  ∈            F              p              ∗      . When    α is a quadratic residue modulo    p, the polynomial        u          2        −  α splits over              Z                      p                  2                     and              R              p      ,      α       is a semi-local ring isomorphic to              Z                      p                  2                      ⊕            Z                      p                  2                    . In this decomposable case, every              R              p      ,      α      -linear cyclic code admits a canonical idempotent decomposition into two cyclic codes over              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              F              p      -linear Gray map is constructed that induces a Lee-to-Hamming isometry from              R              p      ,      α              n       to              F              p              4      n      . Using a compatible bilinear form, we show that the Gray image of a Euclidean self-orthogonal cyclic code remains symplectic self-orthogonal over              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  ↦  −  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              R              p      ,      α       can be used to derive quantum and DNA codes through the Gray map and related algebraic structures.}
}