On the 3/4-Conjecture for Fix-Free Codes -- A Survey

In this survey we concern ourself with the question, wether there exists a fix-free code for a given sequence of codeword lengths. For a given alphabet, we obtain the {\em Kraftsum} of a code, if we divide for every length the number of codewords of this length in the code by the total number of all possible words of this length and then take summation over all codeword lengths which appears in the code. The same way the Kraftsum of a lengths sequence $(l_1,…, l_n) $ is given by $\sum_{i=1}^n q^{-l_i} $, where $q$ is the numbers of letters in the alphabet. Kraft and McMillan have shown in \cite{kraft} (1956), that there exists a prefix-free code with codeword lengths of a certain lengths sequence, if the Kraftsum of the lengths sequence is smaller than or equal to one. Furthermore they have shown, that the converse also holds for all (uniquely decipherable) codes.\footnote{In this survey a code means a set of words, such that any message which is encoded with these words can be uniquely decoded. Therefore we omit in future the “uniquely decipherable” and write only “code”.} The question rises, if Kraft’s and McMillan’s result can be generalized to other types of codes? Throughout, we try to give an answer on this question for the class of fix-free codes. Since any code has Kraftsum smaller than or equal to one, this answers the question for the second implication of Kraft-McMillan’s theorem. Therefore we pay attention mainly to the first implication.

💡 Research Summary

The survey addresses the long‑standing “3/4‑conjecture” for fix‑free codes, a class of codes that are simultaneously prefix‑free and suffix‑free. After recalling the classical Kraft–McMillan theorem, which guarantees the existence of a prefix‑free code whenever the Kraft sum Σ q⁻ˡⁱ ≤ 1, the authors turn to the more restrictive fix‑free setting. They define the Kraft sum for a length sequence (l₁,…,lₙ) as Σ q⁻ˡⁱ, where q is the alphabet size, and note that any code necessarily satisfies this inequality, so the converse direction of Kraft–McMillan is trivial for fix‑free codes. The central question is whether a stronger sufficient condition exists.

The conjecture, first formulated for binary alphabets (q = 2), states that if the Kraft sum is at most 3/4, then a fix‑free code with the prescribed lengths can be constructed. The paper surveys all known partial results supporting the conjecture. Ahlswede, Khachatrian and Litsyn proved it for length sequences with a regular spacing (e.g., lᵢ = i·k). Colbourn and Dinitz showed it holds when the sequence contains only two distinct lengths, by carefully balancing the numbers of short and long codewords. Further work by other authors extended the result to certain “almost regular” distributions and to some families of length sequences generated by linear recurrences. In each case the proof relies on embedding the codewords into a binary tree and demonstrating that the tree’s capacity, measured in terms of available leaves, exceeds the required 3/4 fraction.

Despite these successes, a general proof remains elusive. Counter‑examples are known for Kraft sums strictly between 3/4 and 1: specific length multisets with highly unbalanced short‑word counts cannot be realized as fix‑free codes, showing that the bound 1 is not sufficient in the fix‑free context. The survey discusses these boundary cases and explains why the simple greedy algorithm that works for prefix‑free codes fails when the sum exceeds 3/4. The greedy method places the shortest available codeword in the first free leaf of the tree; when the Kraft sum is larger, the algorithm may become trapped, and the underlying decision problem is known to be NP‑hard.

The authors also examine the situation for non‑binary alphabets. While the 3/4 conjecture is well‑studied for q = 2, very little is known for q ≥ 3. Some heuristic arguments suggest that the admissible Kraft‑sum threshold might increase with q, possibly approaching 1, but no rigorous bound has been established.

Finally, the paper outlines several promising research directions: (1) extending the conjecture to larger alphabets and determining the optimal constant C(q) such that Σ q⁻ˡⁱ ≤ C(q) guarantees a fix‑free code; (2) developing new constructive techniques, for example using Latin‑square‑based placements or probabilistic meta‑heuristics (genetic algorithms, simulated annealing) to explore the space of feasible assignments; (3) investigating the relationships between fix‑free codes and other uniquely decodable families (comma‑free, suffix‑free, etc.) in order to formulate a unified Kraft‑type inequality that encompasses multiple code constraints. The authors conclude that resolving the 3/4‑conjecture, even in the binary case, would not only settle a fundamental combinatorial problem but also provide practical design tools for error‑resilient communication systems.