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In entropy coding, universal coding of integers (UCI) is a binary universal prefix code, such that the ratio of the expected codeword length to <inline-formula> <tex-math notation="LaTeX">$\max \{1, H(P)\}$ </tex-math></inline-formula> is less than or equal to a constant expansion factor <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}$ </tex-math></inline-formula> for any probability distribution <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$H(P)$ </tex-math></inline-formula> is the Shannon entropy of <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula>. <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}^{*}$ </tex-math></inline-formula> is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}^{*}$ </tex-math></inline-formula>. Based on prior research, the range of <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}^{*}$ </tex-math></inline-formula> for the optimal UCI is <inline-formula> <tex-math notation="LaTeX">$2\leq K_{\mathcal {C}}^{*}\leq 2.75$ </tex-math></inline-formula>. Currently, the code constructions achieve <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}=2.75$ </tex-math></inline-formula> for UCI and <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}=3.5$ </tex-math></inline-formula> for asymptotically optimal UCI. In this paper, we construct a class of UCI, termed <inline-formula> <tex-math notation="LaTeX">$\iota $ </tex-math></inline-formula> code, to achieve <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}=2.5$ </tex-math></inline-formula>. This further narrows the range of <inline-formula> <tex-math notation="LaTeX">$K_{\mathcal {C}}^{*}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$2\leq K_{\mathcal {C}}^{*}\leq 2.5$ </tex-math></inline-formula>. Next, a family of asymptotically optimal UCIs is presented, where their expansion factor infinitely approaches 2.5. Then, tighter upper bounds of the expansion factors are derived for several classic UCIs. Finally, we prove that the length of the first codeword of optimal UCI is one.

A Tighter Upper Bound of the Expansion Factor for Universal Coding of Integers and Its Code Constructions