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New upper bounds on the redundancy of Huffman codes are provided. A bound that for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/9 ≤ P₁ ≤ 0.4</tex> is sharper than the bound of Gallager, when the probability of the most likely source letter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P₁</tex> is the only known probability is presented. The improved bound is the tightest possible for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3 ≤ P₁ ≤ 0.4</tex> . Upper bounds are presented on the redundancy of Huffman codes when the extreme probabilities <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P₁</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PN</tex> are known.
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Capocelli et al. (1986) studied this question.
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