On the Linearity and Structure of Z2s-Linear Simplex and MacDonald Codes

摘要

Z2$s$-additive codes are subgroups of Z^{n2s, and can be seen as a generalization of linear codes over Z2and Z4. A Z2$s$ -linear code is a binary code (not necessarily linear) which is the Gray map image of a Z2$s$ -additive code. We consider Z2s- additive simplex codes of type a and β, which are a generalization over $Z$2$s$ of the binary simplex codes. These codes are related to the Z2$s$ -additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding $Z$2s-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z2$s$ -additive MacDonald codes are defined for $s$ > 2, and equivalent results are obtained.}