Systematic encoding for Z₂^s-linear codes

The {\mathbb{Z}_{{2^s}}}-additive codes are subgroups of \mathbb{Z}_{{2^s}}^n. These codes can be seen as a generalization of linear codes over 2 and 4. A {\mathbb{Z}_{{2^s}}}-linear code is a binary code, not necessarily linear, which is the Gray map image of a {\mathbb{Z}_{{2^s}}}-additive code. In 2014, a systematic encoding was found for 4-linear codes. Moreover, an alternative permutation decoding method, which is suitable for any binary code (not necessarily linear) with a systematic encoding, was established. In this paper, we generalise these results by presenting a systematic encoding for {\mathbb{Z}_{{2^s}}}-linear codes with s > 2. This encoding allows us to perform a permutation decoding for this family of codes.