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We demonstrate that there are at least $50$ mutually disjoint Steiner systems $S(5,8,24)$ and there are at least $35$ mutually disjoint $5$-$(24,12,48)$ designs. The latter result provides the existence of a simple $5$-$(24,12,6m)$ design for $m= 24$, 32, 40, 48, 56, 64, 72, 80, 112, 120, 128, 136, 144, 152, 160, 168, 200, 208, 216, 224, 232, 240, 248 and 256.

Mutually Disjoint Steiner Systems $S(5, 8, 24)$ and $5$-$(24,12,48)$ Designs