Generalized Universal Coding of Integers

Universal coding of integers (UCI) is a class of variable-length code such that the ratio of the expected codeword length to
$\max \{1,H(P)\}$
is bounded by a constant factor, where
$H(P)$
is the Shannon entropy of the decreasing probability distribution P. However, if we consider the ratio of the expected codeword length to
$H(P)$
for UCI, the ratio tends to infinity when
$H(P)$
tends to zero. To resolve this issue, we introduce a class of codes, called generalized universal coding of integers (GUCI), where the ratio of the expected codeword length to
$H(P)$
is bounded by a constant factor K. First, the definition of GUCI is proposed. The coding structure of GUCI is introduced. Next, we propose a class of GUCIs
$\mathcal {C}$
to reach the expansion factor
$K_{\mathcal {C}}=2$
, and we show that the smallest minimum expansion factor is in the range
$1\leq K^{*}\leq 2$
. Then, by comparing UCI and GUCI, we show that when the entropy is very large or
$P(0)$
is not large, there are also cases where the average codeword length of GUCI is shorter. Finally, the asymptotically optimal GUCI is presented.