  K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of
unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with
radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the $X$-ray
conjecture and the illumination conjecture for convex bodies of constant width
in $\mathbb{E}^n$, and constructed such coverings for $4\le n\le 6$. Here we
give such constructions with fewer than $2^n$ caps for $5\le n\le 15$.
  For the illumination number of any convex body of constant width in
$\mathbb{E}^n$, O.~Schramm proved an upper estimate with exponential growth of
order $(3/2)^{n/2}$. In particular, that estimate is less than $3\cdot 2^{n-2}$
for $n\ge 16$, confirming the above mentioned conjectures for the class of
convex bodies of constant width. Thus, our result settles the outstanding cases
$7\le n\le 15$.
  We also show how to calculate the covering radius of a given discrete point
set on the sphere efficiently on a computer.
