  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 within 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 $H(P)$, the ratio tends to infinity by using UCI, when
$H(P)$ tends to zero. To solve this issue, this paper introduces a class of
codes, termed generalized universal coding of integers~(GUCI), such that the
ratio of the expected codeword length to $H(P)$ is within a constant factor
$K$. First, the definition of GUCI is proposed and the coding structure of GUCI
is introduced. Next, we propose a class of GUCI $\mathcal{C}$ to achieve the
expansion factor $K_{\mathcal{C}}=2$ and show that the optimal GUCI is in the
range $1\leq K_{\mathcal{C}}^{*}\leq 2$. 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.
