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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.

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Notes

  1. It is described by a more general term down-set in [1].

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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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Authors and Affiliations

  1. Institute of Mathematical Sciences, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul, 03760, Republic of Korea

    Seunghwan Chang

  2. Korea Institute for Advanced Study (KIAS), 85, Hoegiro, Dongdaemun-Gu, Seoul, 02455, Republic of Korea

    Jong Yoon Hyun

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  1. Seunghwan Chang
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  2. Jong Yoon Hyun
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Correspondence to Jong Yoon Hyun.

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Communicated by C. Ding.

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Chang, S., Hyun, J.Y. Linear codes from simplicial complexes. Des. Codes Cryptogr. 86, 2167–2181 (2018). https://doi.org/10.1007/s10623-017-0442-5

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