93% of the ${{ 3}\over { 4}}$ -Conjecture Is Already Verified

One of the most important challenges in the context of fix-free codes is proving the 3/4-conjecture, which guarantees the existence of fix-free codewords of lengths ℓ₁,ℓ₂,..., ℓ_N if Σ_i=1^N2^-ℓi ≤ 3/4. Although this conjecture has not become a theorem yet, some researchers have proved the problem with some extra constraints on the codelengths. One of those, we call it Yekhanin´s constraint, is Σ_{i:ℓi-λ ≤1} 2^-ℓi ≥ 1/2, where λ = minklk. In this paper, it is shown that such a constraint is not so restrictive. We prove that almost 93.8% of the N-tuple codelength vectors with Kraft sum 3/4 do satisfy Yekhanin´s constraint. One can optimistically interpret this result as almost 93.8% of the road of proving the 3/4-conjecture is paved.