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By Norman Abramson

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Information Theory and Coding

Details idea, details and resources, a few houses of Codes, Coding details resources, Channels and Mutual details, trustworthy Messages via Unreliable Channels, thesaurus of Symbols and Expressions.

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The required word lengths, lh h, . . , lq, may or may not be all distinct. We shall find it useful in our construction to consider all words of the same length at one time. Let us, therefore, define rii to be the number of words in our code of length 1; to be the number of words of length 2; etc. If the largest of the k is equal to I, we have i X ni = e (3-8) i = l X 2-<< < 1 (3-5) We may use the in to rewrite (3-7). The summation of (3-7) i=0 By assumption we have lo = 1, h = 2, and h = h — ' ' ' = h = L contains ni terms of the form r"1, nz terms of the form r-2, etc.

Example 4-5. We construct a different code for the source of Example 4-4 in Figure 4-3. 2 binits/symbol and we cannot construct an instantaneous code for this source with a smaller average length. Another point made evident by the synthesis procedure described is that it may sometimes be unnecessary to form a sequence of reductions of the original source all the way to a source with only two symbols. This is so since we need only form reductions until we find the first reduction for which we have a compact code.

A proof of the necessity of the Kraft inequality, on the other hand, cannot be applied to uniquely decodable codes. In fact, the necessary part of the Kraft inequality suggests an investigation of the constraints on the word lengths of uniquely decodable codes. We know that (3-14) expresses a necessary condition for instan¬ taneous codes. Does the same condition hold for the more general uniquely decodable codes? The fact that (3-14) is necessary for uniquely decodable codes, as well as instantaneous codes, was first proved by McMillan (1956).

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