Finely grate the ginger and put it in a pot with the water and sugar and let it boil lightly for 10min.
Pass through a sieve and let it come to room temperature.
Put everything but the carbonated water in a cocktail shaker with ice and shake until it's really cold. This is not 100% needed but it helps in that then the ice in your drink won't melt as it cools it down but rather its job is just to keep the temperature given to the drink in the shaker.
Pour onto a glass with ice and top with carbonated water
Decorate with mint sprigs and lime slices