On information theory parameters of infinite anti-uniform sources

A source S= {s₁, s₂, ...} having a binary Huffman code with code-word lengths satisfying l₁ = 1, l₂ = 2, ... is called an anti-uniform source. If l₁ = 1, l₂ = 2, ... , l_i = i, then the source is called an i-level partially anti-uniform source. The redundancy, expected codeword length and entropy of anti-uniform sources are dealt here. A tight upper bound is derived for the expected codeword length L of anti-uniform sources. It is shown that L does not exceed radic(5 + 3/2). For each 1 < L les (radic(5 + 3)/2), an anti-uniform distribution achieving maximum entropy H(P)_max = L log L - (L - 1)log(L - 1) is introduced. This shows that the maximum entropy achieved by anti-uniform distributions does not exceed 2.512. It is shown that the range of redundancy values for i-level partially anti-uniform sources with distribution {p_i} is an interval of length Sigmaj = i + ₁Pj. This results in a realistic approximation for the redundancy of these sources.