Hi, It sounds like your plants have Cabbage Root Maggot. These are the larvae of a small fly that lays its eggs on the radish tops and the young burrow into the radish itself to eat. The best way to prevent this is to grow your crop under a floating row cover which will keep the flies off of the crop. You can buy floating row covers from Gardener's Supply company, www.gardeners.com. Or, you may be able to find some locally at a well stocked garden center.