In a large bowl, mix together the soy sauce, bean sauce, rice vinegar and 1/2 cup water. In another large bowl, toss the carrots, radishes, cucumber, sugar, salt and ginger.
In a large pot of boiling water, cook the noodles according to package directions, stirring occasionally, until tender but not mushy. Drain and rinse under cold water until the noodles are cool to the touch; shake and toss to drain thoroughly.
Add the noodles to the bowl with the dressing. Add the vegetables, leaving behind any juices, and toss to combine. Divide the salad among 4 plates and top with the scallions.