The original knife bayonet for the Model 1903 was the Model 1905, which replaced the rod bayonet with which the rifle was originally fitted. The Model 1905 bayonet has a 16" blade. That length was chosen because the Krag had a barrel six inches longer and its bayonet had a 10 inch blade. To keep the same bayonet "reach", the new bayonet was made with a 16" blade.
When the M1 rifle was being developed, a cash-poor Army did not want to have to replace all its bayonets, and required that the new rifle use the old bayonet. But meanwhile, tactics had changed, and it was felt that a 10" blade would be adequate, so that length was adopted for any new bayonets (M1 bayonet) and the old Model 1905 bayonets in service were cut down to 10". That situation prevailed through WWII, though a few 16" bayonets escaped cutting and are sometimes seen in WWII photographs.
By the time the M1903A3 was produced, the long M1905 bayonet was obsolete, so the "correct" bayonet for an M1903A3 rifle would be one with a 10" blade, either cut down from a 16" blade or an M1 bayonet made with a 10" blade.