A Tighter Upper Bound of the Expansion Factor for Universal Coding of Integers and Its Code Constructions
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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 max{1, H(P)} is less than or equal to a constant expansion factor K_𝒞 for any probability distribution P, where H(P) is the Shannon entropy of P. K_𝒞^* is the infimum of the set of expansion factors. The optimal UCI is defined as a class of UCI possessing the smallest K_𝒞^*. Based on prior research, the range of K_𝒞^* for the optimal UCI is 2≤ K_𝒞^*≤ 2.75. Currently, the code constructions achieve K_𝒞=2.75 for UCI and K_𝒞=3.5 for asymptotically optimal UCI. In this paper, we propose a class of UCI, termed ι code, to achieve K_𝒞=2.5. This further narrows the range of K_𝒞^* to 2≤ K_𝒞^*≤ 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_𝒞^* for the classic UCIs is discussed.
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