Step 1
Crush 8 oz. ginger, scrubbed, coarsely chopped, with a mortar and pestle or finely chop. Combine ginger, ¾ cup sugar, ¾ tsp. Diamond Crystal or ½ tsp. Morton kosher salt, and ¾ cup water in a medium skillet and bring to a boil. Cook, stirring occasionally, until sugar is dissolved and syrup is slightly reduced, about 3 minutes. Remove from heat and let sit 15 minutes. Step 2
Strain syrup through a fine-mesh sieve into a heatproof measuring glass; discard solids. Stir in ¼ cup fresh lime juice. Step 3
Divide mixture among four 8–10-oz. glasses. Add ice, then top off each glass with about 6 oz. sparkling water. Garnish with lime wedges or wheels.
Do ahead: Ginger syrup (without lime juice) can be made 3 days ahead. Transfer to an airtight container and chill.