Infinite families of $t$-designs from a type of five-weight codes

Abstract: It has been known for a long time that $t$-designs can be employed to construct both linear and nonlinear codes and that the codewords of a fixed weight in a code may hold a $t$-design. While a lot of progress in the direction of constructing codes from $t$-designs has been made, only a small amount of work on the construction of $t$-designs from codes has been done. The objective of this paper is to construct infinite families of $2$-designs and $3$-designs from a type of binary linear codes with five-weights. The total number of $2$-designs and $3$-designs obtained in this paper are exponential in any odd $m$ and the block size of the designs varies in a huge range.