It is not possible to determine the sex of a koi until it is approximately 25cm long because they are not yet sexually mature.
Once a koi has grown longer than 25cm they can be sexed as the testes in males and ovaries in females start to form. The ovaries are a much larger organ than testes and so females are much easier to sex. Females are fairly easy to spot because the belly of the female koi will be plump whereas the male stays fairly streamlined.
When a male is ready for spawning they will develop breeding tubercles on the head and pectoral fins, especially along the bones of the fin rays. These tubercles appear as fine raised spots and could be mistaken for white spot. The tubercle is most visible on the pectoral fins, arranged in fairly regular rows and rough to touch. They are used to in breeding when the male nudges the female with his head and fins to induce her to spawn.