Reduction of Sufficient Number of Code Tables of $k$-Bit Delay Decodable Codes
Kengo Hashimoto, Ken-ichi Iwata
TL;DR
This work addresses reducing the number of code tables needed for $k$-bit delay decodable code-tuples, which can outperform Huffman codes by using a finite set of code tables with decoding delay $k$ and originally required up to $2^{(2^k)}$ tables for optimality. It introduces reduced-code-tuples (RCTs) and a symmetry framework via the set of mappings $\Phi_k$ to partition code-table roles into equivalence classes, proving that optimal $k$-bit delay coding can be achieved using at most one representative per class. A concrete coding procedure is provided that operates on RCTs (without constructing full code-tuples) by pairing class representatives with a small set of symmetry data, yielding the same average codeword length as the corresponding full solution. The results significantly improve practical tractability for theory and implementation, and connect to AIFV-$k$ codes, with remarks on tighter bounds and potential extensions to related coding schemes.
Abstract
A $k$-bit delay decodable code-tuple is a lossless source code that can achieve a smaller average codeword length than Huffman codes by using a finite number of code tables and allowing at most $k$-bit delay for decoding. It is known that there exists a $k$-bit delay decodable code-tuple with at most $2^{(2^k)}$ code tables that attains the optimal average codeword length among all the $k$-bit delay decodable code-tuples for any given i.i.d. source distribution. Namely, it suffices to consider only the code-tuples with at most $2^{(2^k)}$ code tables to accomplish optimality. In this paper, we propose a method to dramatically reduce the number of code tables to be considered in the theoretical analysis, code construction, and coding process.