Combine the carrots, radishes, paprika, and cayenne in a large bowl.
Combine the remaining ingredients in a small saucepan and bring to a boil. Remove from the heat, stirring to dissolve the sugar. Cool to room temperature.
Add the cooled vinegar mixture to the vegetables and stir well. Cover with plastic wrap and let sit at room temperature at least 8 hours or overnight. Refrigerate until ready to serve. (Kim Chee will keep refrigerated in an airtight container for up to 1 week.)
Yield: about 4 cups