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Linear codes from simplicial complexes

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Abstract

In this article we introduce a method of constructing binary linear codes and computing their weights by means of Boolean functions arising from mathematical objects called simplicial complexes. Inspired by Adamaszek (Am Math Mon 122:367–370, 2015) we introduce n-variable generating functions associated with simplicial complexes and derive explicit formulae. Applying the construction (Carlet in Finite Field Appl 13:121–135, 2007; Wadayama in Des Codes Cryptogr 23:23–33, 2001) of binary linear codes to Boolean functions arising from simplicial complexes, we obtain a class of optimal linear codes and a class of minimal linear codes.

Keywords

Simplicial complex Binany linear code Boolean function Walsh–Hadamard transform Optimal linear code Minimal linear code Secret sharing scheme 

Mathematics Subject Classification

94C10 94B05 94A60 

Notes

Acknowledgements

The authors express sincere gratitude to the reviwers for helpful suggestions and comments. The first author was supported by the Ministry of Science, ICT and Future Planning (NRF-2013R1A1A2062121). The second author is supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MEST) (2014R1A1A2A10054745).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of Mathematical SciencesEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Korea Institute for Advanced Study (KIAS)SeoulRepublic of Korea

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