Nonlinearity and kernel of Z2s-linear simplex and MacDonald codes

Abstract

Z2s-additive codes are subgroups of Z2s^n , and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear code is a binary code (not necessarily linear) which is the Gray map image of a Z2s -additive code. We consider Z2s-additive simplex codes of type alpha and beta, which are a generalization over Z2s of the binary simplex codes. These codes are related to the Z2s-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding Z2s-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z2s-additive MacDonald codes are defined for s > 2, and equivalent results are obtained.