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## A general upper bound on the size of constant-weight conflict-avoiding codes

Venue: | IEEE Trans. Inform. Theory |

Citations: | 6 - 6 self |

### Citations

7739 | Matrix Analysis - Horn, Johnson - 1985 |

828 |
A Classical Introduction to Modern Number Theory,
- Ireland, Rosen
- 1990
(Show Context)
Citation Context ...prime, and each of the following: forms a system of distinct representatives of and . Expressed in terms of the Legendre symbol , it is equivalent to (20) and (21) By the law of quadratic reciprocity =-=[7]-=-, (20) and (21) are equivalent to the following conditions: which can be further simplified to or . Hence, for each prime or , we have a - consisting of codewords, which is optimal by Theorem 10. This... |

471 |
An Introduction to the Theory of
- Hardy, Wright
- 1979
(Show Context)
Citation Context ...orem 8 follows by replacing by in Theorem 6. Remark: The celebrated prime number theorem says that approaches 1 when approaches infinity. A weaker form of the prime number theorem proved by Chebyshev =-=[5]-=- states that for some constants and , we can bound by for all . Furthermore, can be upper bounded by for [23]. Hence, for , we have (18) V. OPTIMALITY OF EXISTING CONSTRUCTIONS OF CAC For Hamming weig... |

441 |
Approximate formulas for some functions of prime numbers.
- Rosser, Schoenfeld
- 1962
(Show Context)
Citation Context ...s 1 when approaches infinity. A weaker form of the prime number theorem proved by Chebyshev [5] states that for some constants and , we can bound by for all . Furthermore, can be upper bounded by for =-=[23]-=-. Hence, for , we have (18) V. OPTIMALITY OF EXISTING CONSTRUCTIONS OF CAC For Hamming weight and , Constructions 1 and 2 are shown to be optimal in [16]. In this section, we use the upper bounds on s... |

98 | The collision channel without feedback.
- Massey, Mathys
- 1985
(Show Context)
Citation Context ...ict-avoiding [22] if every subset of sequences out of these sequences is user-irrepressible. User-irrepressible and conflict-avoiding sequences find applications in collision channel without feedback =-=[13]-=-, [21]. In a system with active users, the collision channel is a deterministic channel with inputs and one output defined as follows. Time is assumed to be partitioned into fixed-length time interval... |

65 |
Abschatzung der asymptotischen dichte von summenmengen,
- Kneser
- 1953
(Show Context)
Citation Context ...ANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 7, JULY 2010 IV. UPPER BOUND ON THE SIZE OF CAC In this section we derive an upper bound on the size of CAC. A tool that we will use is Kneser’s theorem =-=[9]-=-, which is a result about the sum of subsets in an abelian group . As we only work with , we will state Kneser’s theorem for . First we introduce some notations. Given two nonempty subsets and of , th... |

32 | Constructions of Binary Constant-Weight Cyclic Codes and Cyclically Permutable Codes
- A, Györfi, et al.
- 1992
(Show Context)
Citation Context ...potential users for a given sequence length. Other coding constructions for multiple access in collision channel without feedback, such as constant-weight cyclically permutable codes, can be found in =-=[1]-=-, [4], [17]. The number of ones in a binary sequence is called the Hamming weight. It is easy to see that in order to support user-irrepressibility, each active user has to send at least packets in a ... |

25 |
Addition theorems: The addition theorems of group theory and number theory
- Mann
- 1965
(Show Context)
Citation Context ...rdinality of the completion of with respective to the stabilizer , plus the cardinality of the completion of with respective to the stabilizer , minus the size of . Proof of Theorem 2 can be found in =-=[12]-=- or [18]. We will apply Kneser’s theorem through the following corollary. Corollary 3: Let be an exceptional codeword in an - and be the stabilizer of , then is periodic, and (4) Proof: Suppose that i... |

17 |
Additive Number Theory–Inverse Problems and the Geometry of Sumsets
- Nathanson
- 1996
(Show Context)
Citation Context ...y of the completion of with respective to the stabilizer , plus the cardinality of the completion of with respective to the stabilizer , minus the size of . Proof of Theorem 2 can be found in [12] or =-=[18]-=-. We will apply Kneser’s theorem through the following corollary. Corollary 3: Let be an exceptional codeword in an - and be the stabilizer of , then is periodic, and (4) Proof: Suppose that is an exc... |

15 | Zinoviev, “New constructions of optimal cyclically permutable constant weight codes - Moreno, Zhang, et al. - 1995 |

14 |
Vajda I.: Constructions of protocol sequences for multiple access collision channel without feedback
- Györfi
- 1993
(Show Context)
Citation Context ...tial users for a given sequence length. Other coding constructions for multiple access in collision channel without feedback, such as constant-weight cyclically permutable codes, can be found in [1], =-=[4]-=-, [17]. The number of ones in a binary sequence is called the Hamming weight. It is easy to see that in order to support user-irrepressibility, each active user has to send at least packets in a perio... |

11 | Design and construction of protocol sequences: Shift invariance and user irrepressibility
- Shum, Wong, et al.
- 2009
(Show Context)
Citation Context ...l for all Hamming weights in general. Index Terms—Conflict-avoiding code, optical orthogonal code, protocol sequence. I. INTRODUCTION A set of binary sequences of length is called user-irre-pressible =-=[20]-=- if after cyclically shifting each of them and stacking them together in a matrix, we can always find a submatrix which is a permutation matrix, regardless of how we shift the sequences. (Recall that ... |

10 | Constant weight conflictavoiding codes
- Momihara, Müller, et al.
- 2007
(Show Context)
Citation Context ...may repeatedly sending the same packet in one sequence period. The packet is guaranteed to be received successfully within the duration of a period. This viewpoint is adopted in [8], [10], [11], [14]–=-=[16]-=-. In this paper, we consider the second design goal and maximize the number of potential users for a given sequence length. Other coding constructions for multiple access in collision channel without ... |

6 |
Conflict-avoiding codes for three active users and cyclic triple systems
- Levenshtein
- 2007
(Show Context)
Citation Context ...ach active may repeatedly sending the same packet in one sequence period. The packet is guaranteed to be received successfully within the duration of a period. This viewpoint is adopted in [8], [10], =-=[11]-=-, [14]–[16]. In this paper, we consider the second design goal and maximize the number of potential users for a given sequence length. Other coding constructions for multiple access in collision chann... |

6 |
Packet communication on a channel without feedback
- Tsybakov, Likhanov
- 1983
(Show Context)
Citation Context ...oiding [22] if every subset of sequences out of these sequences is user-irrepressible. User-irrepressible and conflict-avoiding sequences find applications in collision channel without feedback [13], =-=[21]-=-. In a system with active users, the collision channel is a deterministic channel with inputs and one output defined as follows. Time is assumed to be partitioned into fixed-length time intervals, cal... |

4 |
Conflict-avoiding codes and cyclic triple systems
- Levenshtein
- 2007
(Show Context)
Citation Context ...ted. Each active may repeatedly sending the same packet in one sequence period. The packet is guaranteed to be received successfully within the duration of a period. This viewpoint is adopted in [8], =-=[10]-=-, [11], [14]–[16]. In this paper, we consider the second design goal and maximize the number of potential users for a given sequence length. Other coding constructions for multiple access in collision... |

4 |
A tight asymptotic bound on the size of constant-weight conflict-avoiding codes,” Des
- Shum, Wong
(Show Context)
Citation Context ... address this open question in this paper and provide a general upper bound on the number of potential users for all Hamming weights. An asymptotic version of this general upper bound can be found in =-=[19]-=-. This paper is organized as follows. We define conflict-avoiding codes and set up some notations in Section II. Three known constructions are described in Section III. The main result in this paper i... |

4 |
Rubinov, “Some constructions of conflictavoiding codes
- Tsybakov, R
- 2002
(Show Context)
Citation Context ...e shift the sequences. (Recall that a permutation matrix is a zero-one square matrix with exactly one 1 in each row and each column [6, p. 25].) A set of binary sequences is called -conflict-avoiding =-=[22]-=- if every subset of sequences out of these sequences is user-irrepressible. User-irrepressible and conflict-avoiding sequences find applications in collision channel without feedback [13], [21]. In a ... |

3 |
Øien, “User unsuppressible protocol sequences for collision channel without feedback
- Chen, Shum, et al.
- 2008
(Show Context)
Citation Context ...uences. In the first one, we consider the scenario in which all the users are active, i.e., , and we aim at minimizing the length of the binary sequences while keeping the user-irrepressible property =-=[2]-=-, [20]. We can add an inner code, such as Reed-Solomon code, in order to recover collided packets and enhance system throughput. In the second one, we consider a fixed sequence length and a given numb... |

3 |
Optical orthgonal codes: their bounds and new optimal construction
- Fuji-Hara, Miao
- 2000
(Show Context)
Citation Context ...distinct codewords in an - is no more than 1 for any cyclic shift. An - can thus be viewed as an -optical orthogonal code (OOC) without any autocorrelation requirement. We refer the readers to, e.g., =-=[3]-=-, and the references therein for further information on OOC. A codeword is called equidifference if the elements in form an arithmetic progression in , i.e. for some . In the above equation, the produ... |

1 |
On conflict-avoiding codes of length for three active users
- Jimbo, Mishima, et al.
- 2007
(Show Context)
Citation Context ...upported. Each active may repeatedly sending the same packet in one sequence period. The packet is guaranteed to be received successfully within the duration of a period. This viewpoint is adopted in =-=[8]-=-, [10], [11], [14]–[16]. In this paper, we consider the second design goal and maximize the number of potential users for a given sequence length. Other coding constructions for multiple access in col... |

1 |
Optimal conflict-avoiding codes of length length
- Mishima, Uruno
- 2009
(Show Context)
Citation Context ...tive may repeatedly sending the same packet in one sequence period. The packet is guaranteed to be received successfully within the duration of a period. This viewpoint is adopted in [8], [10], [11], =-=[14]-=-–[16]. In this paper, we consider the second design goal and maximize the number of potential users for a given sequence length. Other coding constructions for multiple access in collision channel wit... |

1 |
and sufficient conditions for tight equidifference conflict-avoiding codes of weight three,” Des
- “Necessary
- 2007
(Show Context)
Citation Context ...imal number of potential users for 0018-9448/$26.00 © 2010 IEEE 3266 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 7, JULY 2010 Hamming weight equal to three, see e.g., [8], [10], [11], [14], =-=[15]-=-. Some optimal constructions for Hamming weight equal to four and five are presented in [16]. However, the maximal number of potential users for general Hamming weight larger than five is unknown. We ... |