Optimality of Huffman Code in the Class of 1-Bit Delay Decodable Codes

Abstract

The class of k-bit delay decodable codes, source codes allowing decoding delay of at most k bits for k ≥ 0, can attain a shorter average codeword length than Huffman codes. This paper discusses the general properties of the class of k-bit delay decodable codes with a finite number of code tables and proves two theorems which enable us to limit the scope of code-tuples to be considered when discussing optimal k-bit delay decodable code-tuples.

Properties of <i>k</i>-Bit Delay Decodable Codes

“…First, we define some notations as follows. Most of the notations in this paper are based on [4]. Let R denote the set of all real numbers, and let R m denote the set of all m dimensional real row vectors for an integer m ≥ 1.…”

“…Definition 5 is equivalent to the definition of k-bit delay decodable codes in [4]. See Appendix A for the proof.…”

“…The literature [4] formalizes a binary encoder with a finite number of code tables as a code-tuple and introduces the class of code-tuples decodable with a delay of at most k bits as the class of k-bit delay decodable codes, which includes the class of AIFV-k codes as a proper subclass. Also, [4] proves that Huffman codes achieve the optimal average codeword length in the class of 1-bit delay decodable code-tuples. The literature [5] indicates that the class of AIFV codes achieves the optimal average codeword length in the class of 2-bit delay decodable code-tuples with two code tables.…”

“…Both theorems enable us to limit the scope of code-tuples to be considered when discussing optimal k-bit delay decodable code-tuples. As an application of these theorems, we can prove of the optimality of AIFV codes in the class of 2-bit delay decodable codes with a finite number of code tables [6]. This paper is organized as follows.…”

General Form of Almost Instantaneous Fixed-to-Variable-Length Codes

“…3. However, it is recently reported [20] that AIFV-2 codes can achieve the optimal expected code length among the codes decodable with 2 bits of decoding delay. This fact means that, for stationary memoryless sources, the modes except {'λ'} and {'01', '1'} are useless in minimizing the code length under the condition of 2-bit decoding delay.…”