Hey, thanks for playing the game! The main reason why I decided to use 4 bits is because I think 3 of them would make the game too easy. Either the controls would be 111 to go up, 000 to go down and 101 to shoot, or they would be a lot more confusing, like 110 to go up, 100 to go down and 101 to shoot.
It also combines with the computer binary digits, that uses 4bits, 8bits, 16bits and so on.