Slice cabbage in half lengthways, remove the core and cut into irregular pieces (about 5cm x 2.5cm/2in x 1in).
Roughly pull pieces apart to separate the leaves.
Slice cucumber and carrots in half lengthways, and then cut into batons (about 6cm x 1cm/2.5in x 0.5in).
Slice daikon radish in half lengthways, and cut into pieces (roughly 3cm x 2cm/1in x 0.5in). Cut red radishes in half.
Place prepared vegetables in a large bowl, sprinkle with salt, and mix well to combine. Cover bowl tightly with plastic wrap and refrigerate overnight.
Combine sugar and vinegar in a heavy-based pot and stir over medium heat until sugar dissolves.
Simmer, without stirring, until reduced and slightly syrupy (about 15 minutes). Set aside to cool.
Next day, pour the cooled syrup over salted vegetables. Add chilli oil and soy sauce to taste, and mix thoroughly.
Store in an airtight container in the refrigerator for three days to allow the flavours to develop before using. The pickles will keep, refrigerated, for several months.