Characterization of Three Dimensional Cellular Automata over $\mathbf{Z}_{m}$

Three dimensional cellular automata wasn't much studied by researches. Tsalides \textit{et al.} characterized three dimensional cellular automata in [1] and then Hemmingsson investigated quasi periodic behavior of three dimensional cellular automata in [2]. In this work we study the algebraic behavior of three dimensional linear cellular automata over $Z_{m}.$ we provide necessary and sufficient conditions for a three dimensional linear cellular automata over the ring $Z_{m}$ to be reversible or irreversible. As a consequence of our result we characterize three dimensional linear cellular automata under the null boundary conditions. \

\textbf{Acknowledgements:} The work is supported by T\"{U}B.{I}TAK (Project Number: 110T713).