@article{Alabiad2025, 
author = {Sami Alabiad and Alhanouf Ali Alhomaidhi},
title = {Various structures of cyclic codes and LCD codes over    G  R  (      p    3    ,  m  )  [  v  ]      /    ⟨      v    2    −      p    2    α  ,  p  v  ⟩},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {11},
pages = {27535-27559},
keywords = {frobenius rings, cyclic codes, self-dual codes, LCD codes classes},
url = {https://www.sciopen.com/article/10.3934/math.20251211},
doi = {10.3934/math.20251211},
abstract = {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  =      G    R    (      p    3    ,  m  )  [  v  ], with        v    2    =      p    2    α,    α  ∈            F                      p        m              ∗  , and    p  v  =  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            +                        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.}
}