A criterion for determining whether multiple shells support a tt-design

In this paper, we provide a criterion for determining whether multiple shells support a $t$-design. We construct as a corollary an infinite series of $2$-designs using power residue codes.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:2309.03206, arXiv:2305.03285, arXiv:2310.1428