My best guess is scab. You can verify this is not being caused by radish root worm by slicing an impacted radish and examining for tunnels and white worm-like animals.
Dry soil conditions favor scab, so irrigate well at planting time. Additionally, do not rotate potatoes with radishes and if necessary, have your garden soil's pH checked by your local extension office. They can assist you in planning how to lower your soil's pH to a level that discourages scab.
Sheridan County Extension
224 South Main, Suite B10
Sheridan, WY 82801-4855
Phone: (307) 674-2980
the "Abstract" section at the top of the page has useful information