We present novel techniques to verify the linearity of$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes, i.e., the binary codes obtained as the image of the generalized Gray map of$\mathbb{Z}_{2^{L}}-\mathbf{additive}$codes. The central idea is the definition of two auxiliary binary codes, which we denote by associated and decomposition codes. Since$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes can be linear or nonlinear, as a consequence of our contributions, we are able to construct families of linear$\mathbb{Z}_{2^{L}}-\mathbf{linear}$codes from nested Reed-Muller and cyclic codes. This work expands on previous results from the literature, where the linearity of$\mathbb{Z}_{2^{L}}. \mathbf{linear}$codes was established with respect to the kernel of the underlying$\mathbb{Z}_{2^{L}}-\mathbf{additive}$code and/or operations on$\mathbb{Z}_{2^{L}}$.