New Binary Fix-Free Codes with Kraft Sum

Two sufficient conditions are given for the existence of binary fix-free codes. The results move closer to the Ahlswede-Balkenhol-Khachatrian conjecture that Kraft sums of at most suffice for the existence of fix-free codes. For each nonnegative integer let denote the set of all binary words of length , and let denote the set of all finite length binary words, including the empty word. A binary code is any finite subset of that does not contain the empty word. The elements of a code are called codewords. For any two words , let denote the concatenation of and . The word is called a prefix of and is called a suffix of . A prefixfree code is a code such that no codeword is a prefix of any other codeword. A suffix-free code is a code such that no codeword is a suffix of any other codeword. A fix-free code is a code that is a both a prefix-free code and a suffix-free code. For any word , let ! #" denote the length of in bits, and let denote the bitwise complement of . If a set $ of numbers if empty, then we adopt the convention %'& ()!*$ ",+.-0/ . In a fix-free code, any finite sequence of codewords can be decoded in both directions, which can improve robustness to channel noise. For any nonnegative mapping 132 405 6 4 , the Kraft sum of 1 is the quantity