-Conjecture for Fix-Free Codes

A fix-free code is a code, which is prefix-free and suffix-free, i.e. any codeword of a fix-free code is neither a prefix, nor a suffix of another codeword. Fix-free codes were first introduced by Schützenberg (4) and Gilbert and Moore (5), where they were called never-self-synchronizing codes. Ahlswede, Balkenhol and Khachatrian propose in (6) the conjecture that a Kraftsum of a lengths sequence smaller than or equal to 3 4 , imply the existence of a fix-free code with codeword lengths of the sequence. This is known as the 3 4 -conjecture for fix-free codes. Harada and Kobayashi generalized in (7) all results of (6) for the case of q-ary alphabets and infinite codes. Over the last years many attempts were done to prove the 34 -conjecture either for the general case of a q-ary alphabet or at least for the special case of a binary alphabet. In this paper we focus mostly on results which shows the 34 -conjecture for special kinds of lengths sequences. The 34 -conjecture holds for finite sequences, if the numbers of codewords on each level is bounded by a term which depends on q and the smallest codeword length which occurs in the lengths sequence. This