In this paper, we construct $3$-designs using extended quadratic residue codes over $\mathbb {F}_q$ and their dual codes. We give as a corollary $3$-designs that do not follow from the transitivity argument and the Assmus--Mattson Theorem.

翻译：本文利用$\mathbb {F}_q$上的扩展二次剩余码及其对偶码构造了$3$-设计。作为推论，我们给出了不依赖于传递性论证和Assmus-Mattson定理的$3$-设计。