Some Results for the Redundancy Bound of the Optimal Ternary AIFV Codes

Ternary AIFV codes are almost instantaneous fixed-to-variable length codes, and are constructed based on two code trees. It is known that the redundancy of ternary AIFV codes is no more than one. In this paper, we provide a tighter upper bound on the redundancy of the ternary Huffman codes when the greatest probability of the source $p_{max}$ is known. As a result, the redundancy of the optimal ternary AIFV codes is bounded by Huffman codes, since the ternary Huffman codes can be seen as the special AIFV codes. To achieve lower redundancy than Huffman codes, we also propose a method to construct a class of ternary AIFV codes with time complexity $O(n)$ for $n$ source symbols. In addition, the redundancy of the proposed AIFV codes is analyzed and compared with Huffman codes. Analyzing the ternary AIFV codes constructed by the algorithm, we derive a tighter redundancy upper bounds under some conditions, which are superior to Huffman codes.