Nonlinearity and Kernel of Z-Linear Simplex and MacDonald Codes

Abstract

Z2s -additive codes are subgroups of Zn2s , 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 α and β, 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.