Improved and Smoothened Upper Bounds on the Redundancy of the Optimal Fix-Free Code

Recently, Yekhanin guaranteed the existence of fix-free codes with codeword lengths (l₁, l₂, ..., l_n) satisfying Sigmaⁿ _i=1 2^-li les 5/8 or Sigmaⁿ _i=1 2^-li) les 3/4 and min _il_i = 1. In this paper, Ye-Yeung approach in deriving upper bound on the redundancy of optimal fix-free code in terms of a known symbol probability q is extended and applied to the new theorems due to Yekhanin. Also, it is shown that for some values of q, assigning a lfloor-log1/2qrfloor bits codeword to the symbol with probability q is preferable to a lceil-log1/2qrceil bits codeword. Noting this point, we remove the discontinuities in the upper bound curves