Current carrying capacity of a cable

The minimum size of the cable to be taken in order to meet the short circuit current carrying capability is dependent on whether the cable is with copper or aluminum conductor. Remember that the k value for copper is calculated to be approximately equal to 143 and and that for aluminium is 94. In both cases the type of insulation is XLPE.

The formula you have indicated is correct.

Also be aware that the way the cable is installed will affect the calculated value e.g. underground in conduit, buried direct, (assuming 3 phase) in trefoil, in thermal insulation etc.

This is true since the ambient temperature will affect the temperature rise of the cable & as such, will affect its fault current carrying capacity as well as its "normal" current carrying capacity.

You will need some standards relevant to your location, from which you can calculate the correct size cable. It cannot be calculated without taking into account the temperature rise of the cable for a given set of installation conditions.