The Arf-Kervaire invariant of framed manifolds as an obstruction to   embeddability
  We prove that no $14$-connected (resp. $30$-connected) stably parallelizable manifold $N^{30}$ (resp. $N^{62}$) of dimension $30$ (resp. $62$) with the Arf-Kervaire invariant 1 can be smoothly embedded into $\mathbb{R}^{36}$ (resp. $\mathbb{R}^{83}$). 
Peter M. Akhmetiev, Matija Cencelj and Du\v{s}an D. Repov\v{s}