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}\over { 4}}$
-conjecture, which guarantees the existence of fix-free codewords of lengths
$\ell _{1},\ell _{2},\ldots, \ell _{N}$
if
$\sum _{i=1}^{N}2^{-\ell _{i}}\leq {{ 3}\over { 4}}$
. 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
$\sum _{i:\ell _{i}-\lambda \leq 1}2^{-\ell _{i}}\geq {{ 1}\over { 2}}$
, where
$\lambda =\min _{k}\ell _{k}$
. 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}\over { 4}}$
do satisfy Yekhanin's constraint. One can optimistically interpret this result as almost 93.8% of the road of proving the
$ {{ 3}\over { 4}}$
-conjecture is paved.