Linear codes over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound

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Abstract

We construct two series of linear codes C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound. Moreover, we consider the Gray images of C(G). The results show that the Gray images of C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) are linear and also reach the Griesmer bound in some cases, and many of linear codes over \(\mathbb {F}_{q}\) we constructed have two Hamming (non-zero) weights.

Keywords

Linear code Galois ring Homogeneous weight Gray map Griesmer bound 

Mathematics Subject Classification

94B05 11T24 
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Notes

Acknowledgements

The work of J. Li was supported by National Natural Science Foundation of China(NSFC) under Grant 11501156 and the Anhui Provincial Natural Science Foundation under Grant 1508085SQA198. The work of A. Zhang was supported by NSFC under Grant 11401468. The work of K. Feng was supported by NSFC under Grant Nos. 11471178, 11571007 and the Tsinghua National Lab. for Information Science and Technology.

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Authors and Affiliations

  1. 1.School of MathematicsHefei University of TechnologyAnhuiChina
  2. 2.Department of Mathematical SciencesXian University of TechnologyShanxiChina
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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