Thoroughly scrub radishes, trim away roots and most of the green stems, leaving 1/8 inch on top and halve or quarter them. Place radishes in a bowl and toss with salt. Let rest for 20 minutes. Drain radishes in a colander set over a bowl, reserving brined juices. Rinse radishes quickly, then shake them to remove excess water.

In a large bowl, stir together 2 tablespoons water with the crushed pepper, ginger, garlic, and sugar. Add drained radishes and mix well to coat with paste. Pack into a pint jar with a tight-fitting lid, then pour the reserved brine into the bowl with the chile paste residue, mix it around to capture leftover seasonings, and pour brine into jar (the liquid will not cover the radishes). Cover and let stand at room temperature overnight. Refrigerate for up to 2 weeks.