Linear codes from simplicial complexes

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.