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Generalized Reed–Muller codes over \({\mathbb{Z}_q}\)

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Abstract

We have given a generalization of Reed–Muller codes over the prime power integer residue ring \({\mathbb{Z}_q}\). These codes are analogs of generalized Reed–Muller (GRM) codes over finite fields. We mainly focus on primitive GRM codes, which are basically a generalization of Quaternary Reed–Muller (QRM) codes. We have also given a multivariate representation of these codes. Non-primitive GRM codes over \({\mathbb{Z}_q}\) are also briefly discussed. It has been shown that GRM codes over \({\mathbb{Z}_q}\) are free extended cyclic codes. A trace description of these codes is also given. We have obtained formulas for their ranks and also obtained expressions for their minimum Hamming distances.

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Authors and Affiliations

  1. Centre for Development of Advanced Computing, B-30, Sector-62, Noida, 201307, India

    Maheshanand Bhaintwal

  2. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India

    Siri Krishan Wasan

Authors
  1. Maheshanand Bhaintwal
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  2. Siri Krishan Wasan
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Correspondence to Maheshanand Bhaintwal.

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Communicated by T. Helleseth.

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Bhaintwal, M., Wasan, S.K. Generalized Reed–Muller codes over \({\mathbb{Z}_q}\) . Des. Codes Cryptogr. 54, 149–166 (2010). https://doi.org/10.1007/s10623-009-9315-x

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  • DOI: https://doi.org/10.1007/s10623-009-9315-x

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