Nested Construction of ℤ2L-Linear Codes

We present novel techniques to verify the linearity of ℤ2L-linear codes, i.e., the binary codes obtained as the image of the generalized Gray map of ℤ2L-additive codes. The central idea is the definition of two auxiliary binary codes, which we denote by associated and decomposition codes. Since ℤ2L-linear codes can be linear or nonlinear, as a consequence of our contributions, we are able to construct families of linear ℤ2L-linear codes from nested Reed-Muller and cyclic codes. This work expands on previous results from the literature, where the linearity of ℤ2L. linear codes was established with respect to the kernel of the underlying ℤ2L-additive code and/or operations on ℤ2L.