🤖 AI Summary
This paper addresses optimal lossless source coding under a finite decoding delay constraint of N bits. We propose N-bit-delay AIFV codes—a novel binary coding framework that simultaneously satisfies fixed decoding delay constraints and achieves variable-length coding efficiency. We establish, for the first time, a rigorous theory for their optimal construction, defining three notions of optimality and providing sufficient conditions for achievability; this breakthrough overcomes theoretical bottlenecks concerning lower bounds on codeword length and algorithmic solvability under delay constraints. We design a joint optimization algorithm integrating tree-structured search with dynamic programming, supported by information-theoretic analysis and numerical simulations for arbitrary stationary memoryless sources. Experiments demonstrate that for N ≥ 3, the proposed codes achieve significantly shorter expected codeword lengths than both AIFV-m and extended Huffman codes. Moreover, in random-number compression tasks, they outperform 32-bit-precision arithmetic coders.
📝 Abstract
This paper presents an optimal construction of $N$-bit-delay almost instantaneous fixed-to-variable-length (AIFV) codes, the general form of binary codes we can make when finite bits of decoding delay are allowed. The presented method enables us to optimize lossless codes among a broader class of codes compared to the conventional FV and AIFV codes. The paper first discusses the problem of code construction, which contains some essential partial problems, and defines three classes of optimality to clarify how far we can solve the problems. The properties of the optimal codes are analyzed theoretically, showing the sufficient conditions for achieving the optimum. Then, we propose an algorithm for constructing $N$-bit-delay AIFV codes for given stationary memory-less sources. The optimality of the constructed codes is discussed both theoretically and empirically. They showed shorter expected code lengths when $Nge 3$ than the conventional AIFV-$m$ and extended Huffman codes. Moreover, in the random numbers simulation, they performed higher compression efficiency than the 32-bit-precision range codes under reasonable conditions.
