In a medium bowl, combine blackberries and sugar. Let stand for 30 minutes.

In the container of a blender, blend blackberry mixture until smooth. Strain mixture through a fine-mesh sieve, discarding solids.

In a gallon-size pitcher, stir together 1½ cups Lime Simple Syrup, sparkling water, and lime juice. Add blackberry mixture, stirring to combine. Taste, and add remaining ¼ cup Lime Simple Syrup, if desired. Serve over ice. Garnish with lime slices and blackberries, if desired.