Combine the garlic, chile and sugar in a mortar and pound with a pestle to a fine paste. Add the lime juice, vinegar, fish sauce and 1/4 cup water. Stir to blend. (Alternately, combine all ingredients in a food processor or blender and process for 30 seconds, until the sugar dissolves. Toss the carrot and daikon shreds with the sugar in a small bowl. Let stand 15 minutes to soften the vegetables. Add the Nuoc Cham to the softened vegetables and stir. On a large platter, decoratively arrange the vegetable ingredients in separate groups.