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