The 3/4 Conjecture for q-Ary Fix-Free Codes With at Most Three Distinct Codeword Lengths

Abstract

We prove the \(3/4\) conjecture for \(q\)-ary fix-free codes with at most three distinct codeword lengths, for every integer \(q\geq2\). Every prescribed length distribution with Kraft sum at most \(3/4\) is realized by a deterministic construction. We introduce a matrix approach based on an exact identity for the overlap between forbidden prefix extensions and suffix residuals. Natural numerical order fixes the shortest layer, and the remaining selection problem is expressed through row and column counts. A fixed-cardinality interpolation theorem supplies feasible sets of every intermediate cardinality between nested endpoints, provided their differences satisfy one-sided uniqueness and either acyclicity or integral slack. Reverse-order selection handles the uniform cases directly; in the remaining cases, layer completion and aligned groups provide endpoints for interpolation. Together, these methods extend the binary three-length result to arbitrary finite alphabets and give a deterministic procedure for constructing the code.