  In entropy coding, universal coding of integers~(UCI) is a binary universal
prefix code, such that the ratio of the expected codeword length to $\max\{1,
H(P)\}$ is less than or equal to a constant expansion factor $K_{\mathcal{C}}$
for any probability distribution $P$, where $H(P)$ is the Shannon entropy of
$P$. $K_{\mathcal{C}}^{*}$ is the infimum of the set of expansion factors. The
optimal UCI is defined as a class of UCI possessing the smallest
$K_{\mathcal{C}}^{*}$. Based on prior research, the range of
$K_{\mathcal{C}}^{*}$ for the optimal UCI is $2\leq K_{\mathcal{C}}^{*}\leq
2.75$. Currently, the code constructions achieve $K_{\mathcal{C}}=2.75$ for UCI
and $K_{\mathcal{C}}=3.5$ for asymptotically optimal UCI. In this paper, we
propose a class of UCI, termed $\iota$ code, to achieve $K_{\mathcal{C}}=2.5$.
This further narrows the range of $K_{\mathcal{C}}^{*}$ to $2\leq
K_{\mathcal{C}}^{*}\leq 2.5$. Next, a family of asymptotically optimal UCIs is
presented, where their expansion factor infinitely approaches $2.5$. Finally, a
more precise range of $K_{\mathcal{C}}^{*}$ for the classic UCIs is discussed.
