Systematic encoding for Z2s-linear codes

The ℤ(2
^s
)-additive codes are subgroups of ℤ
ⁿ
(2
^s
). These codes can be seen as a generalization of linear codes over ℤ
₂
and ℤ
₄
. A ℤ(2
^s
)-linear code is a binary code, not necessarily linear, which is the Gray map image of a ℤ(2
^s
)-additive code. In 2014, a systematic encoding was found for ℤ
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 ℤ(2
^s
)-linear codes with s > 2. This encoding allows us to perform a permutation decoding for this family of codes.