On Optimal and Achievable Fix-Free Codes

Fix-free codes are prefix condition codes which can also be decoded in the reverse direction. They have attracted attention from several communities and are used in video standards. Two variations (with additional constraints) have also been considered for joint source-channel coding: 1) “symmetric” codes, which require the codewords to be palindromes; 2) codes with distance constraints on pairs of codewords. Approaches to determine the existence of a fix-free code with a given set of codeword lengths, for each of the three variations, are proposed. These appear to involve the first use of Boolean satisfiability (SAT) for the existence and design of source codes. Branch-and-bound algorithms to find the collection of optimal codes for asymmetric and symmetric fix-free codes are described. The first bound for the performance of optimal symmetric binary fix-free codes is provided. An earlier conjecture on optimal symmetric binary fix-free codes is proven and related results for asymmetric fix-free codes are presented. A variation of the 3/4 conjecture for fix-free codes is introduced. A key idea in the first conjecture and the new one is a definition of how one sequence of nondecreasing natural numbers dominates another.