  A series of new Cr$_{5+x}$Mo$_{35-x}$W$_{12}$Re$_{35}$Ru$_{13}$C$_{20}$
high-entropy alloys (HEAs) have been synthesized and characterized by x-ray
diffraction, scanning electron microscopy, electrical resistivity, magnetic
susceptibility and specific heat measurements. It is found that the HEAs adopt
a noncentrosymmetric cubic $\beta$-Mn type structure and exhibit bulk
superconductivity for 0 $\leq$ $x$ $\leq$ 9. With increasing $x$, the cubic
lattice parameter decreases from 6.7940(3) {\AA} to 6.7516(3) {\AA}. Meanwhile,
the superconducting transition temperature $T_{\rm c}$ is suppressed from 5.49
K to 3.35 K due to the magnetic pair breaking caused by Cr moments. For all
these noncentrosymmetric HEAs, the zero-temperature upper critical field
$B_{\rm c2}$(0) is comparable to Pauli paramagnetic limit $B_{\rm P}$(0) =
1.86$T_{\rm c}$. In particular, the $B_{\rm c2}$(0)/$B_{\rm P}$(0) ratio
reaches a maximum of $\sim$1.03 at $x$ = 6, which is among the highest for
$\beta$-Mn type superconductors.
