Lay the lime slices on a tray and freeze for at least an hour. Put the remaining chopped limes (including the skin) in the bowl of a food processor or blender with the sugar and 100ml water. Whizz until really juicy and completely broken down.
Pour the mix through a sieve, so that all the pulpy bits remain and you have lots of juice. Have a taste – if it’s too sour, add a bit more sugar. This can be made 2 days ahead and chilled. Pour into a big jug with lots of ice and the lime slices, and top up with soda water. Serve with a shot of vodka or rum, if you like.