A Characterization of $q$-Ary $2$-Designs and a $3$-Design Obstruction for Doubly-Extended Reed-Solomon Codes

Guanghui Zhang, Zhengbang Zha

Let ${\mathcal{C}}_k$ be the doubly extended Reed--Solomon code with parameters $[q+1,k,q-k+2{]}_q$, where $q\ge 3$ is a prime power and $2\le k\le q$. We prove a conjecture of Wu and Z…