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Papers by Holger Schnettler

arXiv (Cornell University), Sep 17, 2007

In this survey we concern ourself with the question, wether there exists a fix-free code for a gi... more 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 (l1,...,ln)(l_1,..., l_n) (l1,...,ln) is given by sumi=1nq−li\sum_{i=1}^n q^{-l_i} sumi=1nq−li, where qqq 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.

ArXiv, 2007

In this survey we concern ourself with the question, wether there exists a fix-free code for a gi... more 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 (l1,...,ln)(l_1,..., l_n) (l1,...,ln) is given by sumi=1nq−li\sum_{i=1}^n q^{-l_i} sumi=1nq−li, where qqq 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 un...

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. On the 3/4-Conjecture for Fix-Free Codes Christian Deppe, Holger Schnettler

Corr, Sep 17, 2007

A Overview of known results about the 3 4-conjecture In Chapter 1 we give first an overview and a... more A Overview of known results about the 3 4-conjecture In Chapter 1 we give first an overview and a proof of the original Kraft-McMillan theorem for prefix-free codes. Then we give a justification of the 3 4conjecture for fix-free codes and examine different forms of the conjecture and the relations among themselves. Especially we show for the general q-ary case that the conjecture holds for 1 2 in place of 3 4 and that for every number bigger than 3 4 the conjecture can not be hold. These theorems were first shown by Ahlswede, Balkenhol and Khachatrian in [5](1996) for the binary case. A generalization was shown by Harada and Kobayashi in [6](1999). Finally we study in Chapter 1 the existence of fix-free extensions of a fix-free code, i.e. we will see, that extensions of fix-free codes are crucially different to extensions of prfix-free codes. Chapter 2 deals with the 3 4-conjecture in the case of a q-ary alphabet. We prove three theorems which show that the conjecture holds for special kinds of lengths sequences. The first theorem occurs first for the binary case in [5](1996) and was generalized in [6](1999). It says, that the conjecture holds, if for two lengths of the sequence, there is a gap of at least twice time of the smaller length, where no other codeword length occur. The second theorem in the chapter shows that the conjecture holds for two level codes and it was proven by Harada and Kobayashi in [5]. Finally we show that the 3 4-conjecture holds for finite sequences, if the numbers of codewords on each level is bounded by a term which depends on q and the smallest codeword length which occurs in the lengths sequence. This theorem was first shown by Kukorelly and Zeger in [10](2003) for the binary case. The generalization of this theorem in Chapter 2 to q-ary alphabets, is one of the new results in this survey. Chapter 3 is a long preparation of Chapter 4. While we will construct fix-free codes from regular subgraphs in the de Bruijn digraph in Chapter 4, we give in Chapter 3 an introduction to the q-ary, n-th level de Bruijn digraph B q (n). Especially we have to know the numbers of vertices, for which there exists a k-regular subgraph in B q (n). De Bruijn graphs were introduced by de Bruijn [29](1946) and Good [30](1946) independently. After a small summary of some basic facts about digraphs and de Bruijn digraphs, we show that for every number L of vertices in B q (n), there exists a cycle of length L in B q (n). This was shown independently by Yoeli, Braynt, Heath , Killick, Golomb, Welch and Goldstein for binary de

2006 IEEE International Symposium on Information Theory, 2006

We treat here the question, whether there exists a q-ary fix-free code for a given sequence of co... more We treat here the question, whether there exists a q-ary fix-free code for a given sequence of codeword lengths. We focus mostly on results which establish the 3/4-conjecture of Ahlswede/Balkenhol/Khachatrian for special classes of lengths sequences. We construct fix-free codes with directed deBrujin graphs. We improve and generalize work of Kukorelly/Zeger and of Yekhanin

Dmtcs Proceedings, Jan 17, 2006

In this paper we concern ourself with the question, whether there exists a fix-free code for a gi... more In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the 3 4-conjecture for special kinds of lengths sequences.

In this paper we concern ourself with the question, whether there exists a fix-free code for a gi... more In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the 3 4-conjecture for special kinds of lengths sequences.

arXiv (Cornell University), Sep 17, 2007

In this survey we concern ourself with the question, wether there exists a fix-free code for a gi... more 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 (l1,...,ln)(l_1,..., l_n) (l1,...,ln) is given by sumi=1nq−li\sum_{i=1}^n q^{-l_i} sumi=1nq−li, where qqq 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.

ArXiv, 2007

In this survey we concern ourself with the question, wether there exists a fix-free code for a gi... more 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 (l1,...,ln)(l_1,..., l_n) (l1,...,ln) is given by sumi=1nq−li\sum_{i=1}^n q^{-l_i} sumi=1nq−li, where qqq 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 un...

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. On the 3/4-Conjecture for Fix-Free Codes Christian Deppe, Holger Schnettler

Corr, Sep 17, 2007

A Overview of known results about the 3 4-conjecture In Chapter 1 we give first an overview and a... more A Overview of known results about the 3 4-conjecture In Chapter 1 we give first an overview and a proof of the original Kraft-McMillan theorem for prefix-free codes. Then we give a justification of the 3 4conjecture for fix-free codes and examine different forms of the conjecture and the relations among themselves. Especially we show for the general q-ary case that the conjecture holds for 1 2 in place of 3 4 and that for every number bigger than 3 4 the conjecture can not be hold. These theorems were first shown by Ahlswede, Balkenhol and Khachatrian in [5](1996) for the binary case. A generalization was shown by Harada and Kobayashi in [6](1999). Finally we study in Chapter 1 the existence of fix-free extensions of a fix-free code, i.e. we will see, that extensions of fix-free codes are crucially different to extensions of prfix-free codes. Chapter 2 deals with the 3 4-conjecture in the case of a q-ary alphabet. We prove three theorems which show that the conjecture holds for special kinds of lengths sequences. The first theorem occurs first for the binary case in [5](1996) and was generalized in [6](1999). It says, that the conjecture holds, if for two lengths of the sequence, there is a gap of at least twice time of the smaller length, where no other codeword length occur. The second theorem in the chapter shows that the conjecture holds for two level codes and it was proven by Harada and Kobayashi in [5]. Finally we show that the 3 4-conjecture holds for finite sequences, if the numbers of codewords on each level is bounded by a term which depends on q and the smallest codeword length which occurs in the lengths sequence. This theorem was first shown by Kukorelly and Zeger in [10](2003) for the binary case. The generalization of this theorem in Chapter 2 to q-ary alphabets, is one of the new results in this survey. Chapter 3 is a long preparation of Chapter 4. While we will construct fix-free codes from regular subgraphs in the de Bruijn digraph in Chapter 4, we give in Chapter 3 an introduction to the q-ary, n-th level de Bruijn digraph B q (n). Especially we have to know the numbers of vertices, for which there exists a k-regular subgraph in B q (n). De Bruijn graphs were introduced by de Bruijn [29](1946) and Good [30](1946) independently. After a small summary of some basic facts about digraphs and de Bruijn digraphs, we show that for every number L of vertices in B q (n), there exists a cycle of length L in B q (n). This was shown independently by Yoeli, Braynt, Heath , Killick, Golomb, Welch and Goldstein for binary de

2006 IEEE International Symposium on Information Theory, 2006

We treat here the question, whether there exists a q-ary fix-free code for a given sequence of co... more We treat here the question, whether there exists a q-ary fix-free code for a given sequence of codeword lengths. We focus mostly on results which establish the 3/4-conjecture of Ahlswede/Balkenhol/Khachatrian for special classes of lengths sequences. We construct fix-free codes with directed deBrujin graphs. We improve and generalize work of Kukorelly/Zeger and of Yekhanin

Dmtcs Proceedings, Jan 17, 2006

In this paper we concern ourself with the question, whether there exists a fix-free code for a gi... more In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the 3 4-conjecture for special kinds of lengths sequences.

In this paper we concern ourself with the question, whether there exists a fix-free code for a gi... more In this paper we concern ourself with the question, whether there exists a fix-free code for a given sequence of codeword lengths. We focus mostly on results which shows the 3 4-conjecture for special kinds of lengths sequences.