Linear codes from simplicial complexes
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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.
KeywordsSimplicial complex Binany linear code Boolean function Walsh–Hadamard transform Optimal linear code Minimal linear code Secret sharing scheme
Mathematics Subject Classification94C10 94B05 94A60
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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