AIFV (almost instantaneous fixed-to-variable length) codes are noiseless source codes that can attain a shorter average codeword length than Huffman codes by allowing a time-variant encoder with two code tables and a decoding delay of at most 2 bits. First, we consider a general class of noiseless source codes, called k-bit delay decodable codes, in which one allows a finite number of code tables and a decoding delay of at most k bits for k >= 0. Then we prove that AIFV codes achieve the optimal average codeword length in the 2-bit delay decodable codes class.