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

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Abstract

In an earlier paper the authors studied simplex codes of type α 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 α 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.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Ohio State University, Columbus, OH, 43210, USA

    Manish K. Gupta

  2. Department of Mathematics, Indian Institute of Technology, Kanpur, India

    Mahesh C. Bhandari & Arbind K. Lal

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  1. Manish K. Gupta
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  2. Mahesh C. Bhandari
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  3. Arbind K. Lal
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Correspondence to Manish K. Gupta.

Additional information

A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, India

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Gupta, M.K., Bhandari, M.C. & Lal, A.K. On Linear Codes over \({\mathbb{Z}}_{2^{s}}\). Des Codes Crypt 36, 227–244 (2005). https://doi.org/10.1007/s10623-004-1717-1

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