Linear codes and $t$-designs are interactive with each other. It is well known that some $t$-designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended 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 extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold $3$-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal.

翻译：线性代码和美元设计是互相互动的,众所周知,近年来某些美元设计是通过使用某些线性代码建造的,然而,文献中只有少量的线性代码扩展代码的无限家庭,拥有价值3美元的无限家庭设计价值为$t$Geq 3美元。在本文中,我们研究了具有3个零及其双重代码的二元周期代码强化代码的扩展代码,并表明这些代码持有价值3美元的设计。 此外,我们还从研究过的周期代码中获得了一些缩短的代码,并明确确定了其参数。 其中一些缩短的代码是最佳或几乎最理想的。