论文标题
大约3美元的设计$ 3 $ - 带有三个零的二进制循环代码的代码
Some $3$-designs and shortened codes from binary cyclic codes with three zeros
论文作者
论文摘要
线性代码和$ t $ -Designs相互交互。众所周知，近年来使用某些线性代码来构建了一些$ t $ designs。但是，文献中报告了只有少数数量的无限码线性代码的线性代码，其中包含一个无限的家族，其中包括$ t $ designs $ t \ geq 3 $。在本文中，我们研究了带有三个零及其双重代码的一类二进制循环代码的增强代码的扩展代码，并表明这些代码容纳$ 3 $ designs。此外，我们从研究的循环代码中获得了一些缩短的代码，并明确确定其参数。这些缩短的代码中的一些是最佳或几乎最佳的。
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.
