A lot of the growth depends on the genetics of the fish. As Koi are ornamental variants of
Cyprinus carpio the same principles should apply in the majority of cases. In the angling world, there are strains of
C. carpio that grow very fast (can reach 40lb+ in not much more than 6 years!) but these tend to have short lifespans in comparison to traditional strains that can take anywhere up to 20-30 years to reach 30lb in weight (dependant on food sources and quality of nutrition from them). If you want the fish to grow quickly then a high nutritional value protein/carbohydrate food will promote growth, but must also have amino acids and vitamins in the food. Angling baits of this nature cost on average £10 per kilo, and I do feed my fish occasionally with this type of food source (ground down to small pieces rather than the 16mm diameter "balls" of bait), but I know exactly what is in the food due to working with the bait company to develop it (has high amounts of natural ingredients such as real krill, squid meal and crab meat backed up with an amino acid profile and vitamins). The only drawback is that it sinks so has to be fed sparingly and watched as the fish consume it so not to overfeed (although it does breakdown quickly over a few days in a natural lake).
I don't know much about the koi strains, but the strain of
C. carpio I have in my pond are not known for their fast growth, but are renowned for living a long time and being absolutely gorgeous looking fish (dark colouration with almost perfect scale patterns along the lateral line). I should have taken some pics when I transferred them earlier to the new pond, but didn't think about it.
I have attached 4 pics to this post to show fish that are all over 25 years old and from the "Leney" strain descendancy (from 2 different pools - one in Devon and the other in Warwickshire). All are between 21lb & 25lb in weight from two very differently stocked waters (the heavy vegetation background is an acre in size and holds over 150 15lb+ fish!!! where as the bare scenery pool is again an acre in size but only holds 30 carp but has a very high amount of smaller cyprinids competing for the food). In the last picture, the scale was lost during spawning last year, but other than being light in colour there, no infection or other issues are present (I treated it with an antiseptic solution which was then secured with propolis).
For me it is not about the size, but the genetic strain of the fish (in angling circles known as heritage or history fish).