Ice cubes: Cut each lime slice in quarters and put them into 16 ice cube molds. Combine the water and lime juice and pour it over the lime slices. Freeze until frozen, at least 4 hours.
Syrup: In a small saucepan, combine the water, sugar, and ginger over medium heat. Bring to a boil, then reduce the heat and simmer, stirring occasionally until the sugar has dissolved, about 5 minutes. Remove the pan from the heat and allow the mixture to cool, about 20 minutes. Cover the pan and refrigerate for 4 hours.
Chill 4 martini glasses in the freezer. Fill a cocktail shaker with ice. Add the vodka, ginger syrup and soda water. Shake for 10 seconds, then strain the mixture into the prepared glasses. Put 1 to 2 lime-flavored ice cubes in each glass and serve.