A Complete Characterization of q-Ary $2$-Designs in Doubly-Extended Reed-Solomon Codes

Let Cₖ be the doubly-extended Reed--Solomon code with parameters [q+1, k, q-k+2]q. We give a complete characterization of when its codewords of a fixed weight form q-ary 2-designs. The main ingredient is an exact criterion showing that the number of r-subsets of Fq^* with a prescribed product is independent of that product if and only if (r, q-1)=1. Applying this criterion to the minimum-weight codewords of Cₖ, we prove that they form a q-ary 2-design if and only if (k-1, q-1)=1. We further establish a propagation result for MDS codes showing that the q-ary 2-design property at minimum weight implies the same property at every nonzero weight. Consequently, Cₖ holds q-ary 2-designs at all nonzero weights if and only if (k-1, q-1)=1.