  An empty simplex is a lattice simplex in which vertices are the only lattice
points. We show two constructions leading to the first known empty simplices of
width larger than their dimension:
  - We introduce cyclotomic simplices and exhaustively compute all the
cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we
find five empty ones of width $11$, and none of larger width.
  - Using circulant matrices of a very specific form, we construct empty
simplices of arbitrary dimension $d$ and width growing asymptotically as
$d/\operatorname{arcsinh}(1) \sim 1.1346\,d$.
  The latter are only $0.3\%$ less wide than the widest hollow convex bodies
known.
