By Norman Abramson
Info idea, details and assets, a few houses of Codes, Coding details assets, Channels and Mutual details, trustworthy Messages via Unreliable Channels, word list of Symbols and Expressions.
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Are provided by the Kraft inequality (Kraft, 1949). A necessary and sufficient condition for the existence of an instantaneous code with word lengths h, U, ... ,lq is that si —> 00 I r~li *< 1 Then we may set S2—>01 and we still have two prefixes of length 2 which are unused. We may use these as shown on the next page. 4=1 where r is the number of different symbols in the code alphabet. (3-2) 54 INFORMATION THEORY AND CODING SOME PROPERTIES OF CODES 55 For the binary case, the Kraft inequality tells us that the U must satisfy the equation this ease it is clear not only that the word lengths of code a satisfy e Kraft inequality, but also that the words themselves do com¬ prise an instantaneous code.
Equations (4-7) express a bound for the average length L, independent of any particular coding scheme. The bound requires only that our code be instantaneous. Equation (4-12), on the other Equation (4-15a) is known as Shannon's first theorem or the noiseless coding theorem. It is one of the two major theorems of information theory. Equation (4~15a) tells us that we can make the average number of r-ary code symbols per source symbol as small as, but no smaller than, the entropy of the source measured in r-ary units.
It makes no difference which of the two words formed is assigned to which of the source symbols. This means that the assignment of the two code symbols 0 and 1 to the various words of the compact code we construct is arbitrary. We may complement f the jih digit of every word of the code and obtain another compact code. For example, if we complement the first and last digits of the code of Figure 4-2, we have the "new" compact code: f The complement of 0 is 1; the complement of 1 is 0. 2 binits/symbol This method of producing a new compact code, however, results in just trivial differences in the two codes.