Kraft's Inequality - Proof of The General Case

Proof of The General Case

Consider the generating function in inverse of x for the code S

in which —the coefficient in front of —is the number of distinct codewords of length . Here min is the length of the shortest codeword in S, and max is the length of the longest codeword in S.

For any positive integer m consider the m-fold product Sm, which consists of all the words of the form, where are indices between 1 and n. Note that, since S was assumed to uniquely decodable, if, then . In other words, every word in comes from a unique sequence of codewords in . Because of this property, one can compute the generating function for from the generating function as

G(x) = \left( F(x) \right)^m = \left( \sum_{i=1}^n x^{-|s_i|} \right)^m
= \sum_{i_1=1}^n \sum_{i_2=1}^n \cdots \sum_{i_m=1}^n x^{-\left(|s_{i_1}| + |s_{i_2}| + \cdots + |s_{i_m}|\right)} = \sum_{i_1=1}^n \sum_{i_2=1}^n \cdots \sum_{i_m=1}^n x^{-|s_{i_1} s_{i_2}\cdots s_{i_m}|}
= \sum_{\ell=m \cdot \min}^{m \cdot \max} q_\ell \, x^{-\ell} \; .

Here, similarly as before, —the coefficient in front of in —is the number of words of length in . Clearly, cannot exceed . Hence for any positive x


\left( F(x) \right)^m \le \sum_{\ell=m \cdot \min}^{m \cdot \max} r^\ell \, x^{-\ell} \; .

Substituting the value x = r we have


\left( F(r) \right)^m \le m \cdot (\max-\min)+1

for any positive integer . The left side of the inequality grows exponentially in and the right side only linearly. The only possibility for the inequality to be valid for all is that . Looking back on the definition of we finally get the inequality.


\sum_{i=1}^n r^{-\ell_i} = \sum_{i=1}^n r^{-|s_i|} = F(r) \le 1 \; .

Read more about this topic:  Kraft's Inequality

Famous quotes containing the words proof of the, proof of, proof, general and/or case:

    From whichever angle one looks at it, the application of racial theories remains a striking proof of the lowered demands of public opinion upon the purity of critical judgment.
    Johan Huizinga (1872–1945)

    The fact that several men were able to become infatuated with that latrine is truly the proof of the decline of the men of this century.
    Charles Baudelaire (1821–1867)

    The thing with Catholicism, the same as all religions, is that it teaches what should be, which seems rather incorrect. This is “what should be.” Now, if you’re taught to live up to a “what should be” that never existed—only an occult superstition, no proof of this “should be”Mthen you can sit on a jury and indict easily, you can cast the first stone, you can burn Adolf Eichmann, like that!
    Lenny Bruce (1925–1966)

    That sort of half sigh, which, accompanied by two or three slight nods of the head, is pity’s small change in general society.
    Charles Dickens (1812–1870)

    As one knows the poet by his fine music, so one can recognise the liar by his rich rhythmic utterance, and in neither case will the casual inspiration of the moment suffice. Here, as elsewhere, practice must precede perfection.
    Oscar Wilde (1854–1900)