## Combinatorial problems motivated by comma-free codes (2004)

Venue: | J. Combin. Des |

Citations: | 3 - 0 self |

### BibTeX

@ARTICLE{Levenshtein04combinatorialproblems,

author = {Vladimir I. Levenshtein},

title = {Combinatorial problems motivated by comma-free codes},

journal = {J. Combin. Des},

year = {2004},

volume = {12},

pages = {184--196}

}

### OpenURL

### Abstract

Abstract In the paper some combinatorial problems motivated by comma-free codes are considered. We describe these problems, give the most significant known results and methods used, present some new results and formulate open problems.

### Citations

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String overlaps, pattern matching, and non-transitive games
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(Show Context)
Citation Context ... for n ≥ k+2 all words of G k,m q (n) begin with a prefix of k+1 nonzero letters allows us to apply Theorem 2.1 (after correcting some misprints in its formulation) of the paper by Guibas and Odlyzko =-=[11]-=- to find a generating function f − (G k,m q , z) = � n=k+2 We give the result for q = 2 and k = 1. Theorem 12 f − (G 1,m |G k,m q (n)|z −n . 2 , z) = �m−1 i=1 xi where x0, x1, ..., xm−1 is a solution ... |

22 |
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Citation Context ...contains elements of H only in the first and last m positions. Note that some words of CH can have length smaller than m. Codes with one synchronizing suffix (|H| = 1) were investigated by Gilbert in =-=[7]-=-. In the case of suffix 0m these are codes without overlaps � F n q and G(n, q) = which we denote by G 0,m q . Let G 0,m q (n) = G 0,m q max |G 0,m q (n)| where the maximum is taken over all m ≥ 1. Th... |

19 |
Variable-length binary encodings,” Bell Syst
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(Show Context)
Citation Context ...xists an integer s such that knowledge of s consecutive letters of any sequence of codewords allows one to uniquely determine at least one place (comma) separating two codewords. Gilbert and Moore in =-=[6]-=- considered the property of bounded decoding delay of variable length codes. This property suggests the existence of a decoding automaton or a machine with a finite number of states which, starting fr... |

13 |
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Citation Context ...ew direction in coding theory: investigation of combinatorial problems of word synchronization for block codes. Although commafree codes were introduced one year earlier by Crick, Griffith, and Orgel =-=[3]-=-, paper [8] contains essential mathematical results for these codes: like for example an upper bound on the codes’ size and a proof that, for any alphabet of size q, this bound is tight when n, the le... |

12 |
L.R.: Comma-free codes
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(Show Context)
Citation Context ...late open problems. Keywords: comma-free codes, comma-free index, comma-free sequence, codes without overlaps, difference systems of sets, synchronization 1. Introduction The paper “Comma-free codes” =-=[8]-=- by Golomb, Gordon, and Welch opened a new direction in coding theory: investigation of combinatorial problems of word synchronization for block codes. Although commafree codes were introduced one yea... |

9 |
L.: On the construction of comma-free codes
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Citation Context ...he length of the codewords, is odd and sufficiently small. Numerous efforts were overtaken to prove that this bound is tight for each odd length and this task was successfully completed by Eastman in =-=[5]-=-. On the other hand, it was proved in [8] that this bound cannot be attained for even n if q is large compared to n. The effort to strengthen Eastman’s result gave rise [12], [22] to an investigation ... |

8 |
Codes with bounded synchronization delay
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(Show Context)
Citation Context ...s rise to the notion of bounded synchronization delay of variable length codes. This notion was independently introduced and investigated by the author in [13], [14], [15] and by Golomb and Gordon in =-=[10]-=-. In this case there exists an integer s such that knowledge of s consecutive letters of any sequence of codewords allows one to uniquely determine at least one place (comma) separating two codewords.... |

8 |
Recent results in comma-free codes
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Citation Context ...lly completed by Eastman in [5]. On the other hand, it was proved in [8] that this bound cannot be attained for even n if q is large compared to n. The effort to strengthen Eastman’s result gave rise =-=[12]-=-, [22] to an investigation of a very interesting combinatorial problem on the maximum number of pairwise comparable vectors of length n over the alphabet {0, 1, ∗}. Section 2 contains the main results... |

4 |
On the maximum number of words in codes without overlaps
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(Show Context)
Citation Context ... automaton. Such an automaton possesses an additional non-error propagation property: its initialization in any state can result in wrong decoding of a bounded number t of initial codewords. In [16], =-=[18]-=-, the author introduced and investigated codes without overlaps whose characteristic property is the existence of decoding automaton invariant with respect to the initial state (or self-adjusting deco... |

4 |
0,1,*)-Distance Problems i.n Combinatorics
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Citation Context ...r x (a) r > x (b) r or x (a) r < x (b) r ; converse inequality on another position s is not possible by the main argument above. Results on t(l) presently known are described in the paper by van Lint =-=[23]-=- with full proofs. An upper bound t(l) < l c log 2 l with a constant c > 1 was proved in [22]. Using the known recurrence t(l) ≤ t(l − 1) + t(⌊ (6) l − 1 ⌋) (7) 2 van Lint [23] proved (6) with c = 1. ... |

3 |
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Citation Context ...mpleted by Eastman in [5]. On the other hand, it was proved in [8] that this bound cannot be attained for even n if q is large compared to n. The effort to strengthen Eastman’s result gave rise [12], =-=[22]-=- to an investigation of a very interesting combinatorial problem on the maximum number of pairwise comparable vectors of length n over the alphabet {0, 1, ∗}. Section 2 contains the main results for c... |

2 | Construction and properties of comma-free codes - Golomb, Welch, et al. - 1958 |

2 |
Maximal and variable word-length comma-free codes
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(Show Context)
Citation Context ... 2 (Eastman [5], 1965) For any q ≥ 2 and any odd n E n q is a comma-free code and |E n q | = B(n, q). (3) Another construction of maximum comma-free codes of odd length was presented later by Scholtz =-=[21]-=-. His construction was iterative and applicable to codes of variable length. In the meantime in [8] it was noted that, in general, (2) is not tight for even n and it was proved that M(2, q) = ⌊ 1 3 q2... |

1 |
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Citation Context ...nd n → ∞. The high encoding and decoding complexity of these codes makes the investigation of the comma-free index of cosets of linear [n, k] codes of essential interest. However, Bassalygo proved in =-=[1]-=- that the redundancy r = n − k of such codes with comma-free index ρ must be at least √ ρn. For finding cosets of linear codes whose redundancy is close to this bound, the author introduced and consid... |

1 |
Contemporary Design Theory: A
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(Show Context)
Citation Context ...) (or DSS-(v, 1, k, 1)) are known to exist for parameters k = t + 1 and v = t 2 + t + 1 where t is a prime power. There exists a very possible conjecture that these are the only valid parameters (see =-=[4]-=-). This implies that rt+1(t 2 + t + 1, 1) ≤ t + 1 if t is a prime power. Note that from (17) it follows that rq(n, 1) � √ n as q and n → ∞. (20) Thus, for the known cyclic difference sets, the asympto... |

1 |
properties of code systems”, Dokl
- Levenshtein, “Certain
- 1961
(Show Context)
Citation Context ...onsideration of (block) comma-free codes gives rise to the notion of bounded synchronization delay of variable length codes. This notion was independently introduced and investigated by the author in =-=[13]-=-, [14], [15] and by Golomb and Gordon in [10]. In this case there exists an integer s such that knowledge of s consecutive letters of any sequence of codewords allows one to uniquely determine at leas... |

1 |
Self-adaptive automata for message decoding”, Dokl
- Levenshtein
- 1961
(Show Context)
Citation Context ...ration of (block) comma-free codes gives rise to the notion of bounded synchronization delay of variable length codes. This notion was independently introduced and investigated by the author in [13], =-=[14]-=-, [15] and by Golomb and Gordon in [10]. In this case there exists an integer s such that knowledge of s consecutive letters of any sequence of codewords allows one to uniquely determine at least one ... |

1 |
properties of code systems and self-adjusting automata for message decoding
- Levenshtein, “Certain
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(Show Context)
Citation Context ... of (block) comma-free codes gives rise to the notion of bounded synchronization delay of variable length codes. This notion was independently introduced and investigated by the author in [13], [14], =-=[15]-=- and by Golomb and Gordon in [10]. In this case there exists an integer s such that knowledge of s consecutive letters of any sequence of codewords allows one to uniquely determine at least one place ... |

1 |
Decoding automata which are invariant with respect to their initial state
- Levenshtein
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(Show Context)
Citation Context ...coding automaton. Such an automaton possesses an additional non-error propagation property: its initialization in any state can result in wrong decoding of a bounded number t of initial codewords. In =-=[16]-=-, [18], the author introduced and investigated codes without overlaps whose characteristic property is the existence of decoding automaton invariant with respect to the initial state (or self-adjustin... |

1 |
Bounds for codes ensuring error correction and synchronization”, Problemy Peredachi
- Levenshtein
- 1969
(Show Context)
Citation Context ...0023241. 1s2 The natural generalization of comma-free codes to codes with given comma-free index or code separation was caused by considering word synchronization in erroneous channels. The author in =-=[17]-=- showed how it is possible to combine error-correcting and word-synchronization properties. In particular, there exist binary codes C of length n with minimum Hamming distance 2s + 1 and comma-free in... |

1 |
One method of constructing quasilinear codes providing synchronization in the presence of errors”, Problemy Peredachi
- Levenshtein
- 1971
(Show Context)
Citation Context ...e redundancy r = n − k of such codes with comma-free index ρ must be at least √ ρn. For finding cosets of linear codes whose redundancy is close to this bound, the author introduced and considered in =-=[19]-=- a combinatorial problem on the difference systems of sets. The problem is to find a sequence (or a code of size 1) of length n over the alphabet {0, 1, ..., q − 1, ∗} which has a comma-free index ρ a... |

1 |
Construction and properties of binary constantweight codes without overlaps
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- 1971
(Show Context)
Citation Context .... . q − 1 (24) Let F n,w 2 be the subset of all words of F n 2 which have w ones. For any relatively prime n and w, a maximum code C ⊂ F n,w 2 without overlaps was constructed by Markov and Noskov in =-=[20]-=-. Their construction is based on the remarkable fact that every cyclic class of words of F n,w 2 contains a unique representative x ∈ F n,w 2 such that for any of its prefices y ∈ F n′ ,w ′ 2 , 1 ≤ n ... |