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Linear codes and $t$-designs are interactive with each other. It is wellknown that some $t$-designs have been constructed by using certain linear codesin recent years. However, only a small number of infinite families of theextended codes of linear codes holding an infinite family of $t$-designs with$t\geq 3$ are reported in the literature. In this paper, we study the extendedcodes of the augmented codes of a class of binary cyclic codes with three zerosand their dual codes, and show that those codes hold $3$-designs. Furthermore,we obtain some shortened codes from the studied cyclic codes and explicitlydetermine their parameters. Some of those shortened codes are optimal or almostoptimal.

Some $3$-designs and shortened codes from binary cyclic codes with three zeros