关于Fix-Free码的3/4猜想--综述
On the 3/4-Conjecture for Fix-Free Codes -- A Survey
摘要：
在这篇综述中，我们关心的问题是，对于给定的码字长度序列，是否存在一个固定的自由码。对于一个给定的字母表，我们得到的{ \em Kraftsum}的代码，如果我们除以这个长度的码字在代码中的所有可能的这个长度的字的总数，然后采取所有码字长度的总和出现在代码中。同样，长度序列$(l_1,..., l_n) $的Kraftsum由$\sum_{i=1}^n q^{-l_i} $给出，其中$q$是字母表中字母的数量。Kraft和McMillan在\cite{kraft}（1956）中指出，如果长度序列的Kraftsum小于或等于1，则存在具有一定长度序列的码字长度的无前缀码。此外，他们还表明，逆也适用于所有（唯一可破译的）代码。\footnote{In this survey a code means a set of words, such that any message which is encoded with these words can be uniquely decoded. Therefore we omit in future the "uniquely decipherable" and write only "code".}问题来了，如果卡夫和麦克米兰的结果可以推广到其他类型的代码？在整个过程中，我们试图给出一个答案，这个问题的类fix-free码。由于任何代码的Kraftsum都小于或等于1，这就回答了Kraft-McMillan定理的第二个含义的问题。因此，我们主要关注第一个含义。
英文摘要：
In this survey we concern ourself with the question, wether there exists a fix-free code for a given sequence of codeword lengths. For a given alphabet, we obtain the {\em Kraftsum} of a code, if we divide for every length the number of codewords of this length in the code by the total number of all possible words of this length and then take summation over all codeword lengths which appears in the code. The same way the Kraftsum of a lengths sequence $(l_1,..., l_n) $ is given by $\sum_{i=1}^n q^{-l_i} $, where $q$ is the numbers of letters in the alphabet. Kraft and McMillan have shown in \cite{kraft} (1956), that there exists a prefix-free code with codeword lengths of a certain lengths sequence, if the Kraftsum of the lengths sequence is smaller than or equal to one. Furthermore they have shown, that the converse also holds for all (uniquely decipherable) codes.\footnote{In this survey a code means a set of words, such that any message which is encoded with these words can be uniquely decoded. Therefore we omit in future the "uniquely decipherable" and write only "code".} The question rises, if Kraft's and McMillan's result can be generalized to other types of codes? Throughout, we try to give an answer on this question for the class of fix-free codes. Since any code has Kraftsum smaller than or equal to one, this answers the question for the second implication of Kraft-McMillan's theorem. Therefore we pay attention mainly to the first implication.
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