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The linear codes of t-designs held in the Reed-Muller and Simplex codes (2008.09935v1)
Published 23 Aug 2020 in cs.IT and math.IT
Abstract: A fascinating topic of combinatorics is $t$-designs, which have a very long history. The incidence matrix of a $t$-design generates a linear code over GF$(q)$ for any prime power $q$, which is called the linear code of the $t$-design over GF$(q)$. On the other hand, some linear codes hold $t$-designs for some $t \geq 1$. The purpose of this paper is to study the linear codes of some $t$-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of $t$-designs held in linear codes is presented. Open problems are also presented.