On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$

Abstract

In an earlier paper the authors studied simplex codes of type ? and β over $${\mathbb{Z}}_4$$ and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type ? and β over $${\mathbb{Z}}_{{2^s}}.$$ The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over $${\mathbb{Z}}_{2^{s}}.$$ The above codes are also shown to satisfy the chain condition.