ICT12 2. Huffman codes

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ICT12 2. Huffman codes

10To prove the second part, let l j be the least integer such that r l j≥ 1/p j .Then p j ≥ r −l jand ∑ r −l jj ≤ ∑ j p j = 1. Thus the l j satisfy the Kraft-McMillan inequality and hence there is a prefix (sic) encoding f: A → C ∗with code lengths l 1 , … , l n . But l j < 1 − log 2 p j / log 2 (r), by E.2.2, andhence l = ∑ j p j l j< 1 + H/log 2 (r).E.2.2. Show that the least integer such that r l j≥ 1/p j is− log 2 p j / log 2 (r).In particular, l j < 1 − log 2 p j / log 2 (r).E.2.3. Use the Kraft-McMillan inequality to show that the setC = {0, 01, 11, 101}is not a code and find a binary string that can be decomposed in two differentways as a concatenation of elements of C.

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