With the challenge of quantum computing ahead, an analysis of number and representation adequate to the task is needed. Some clarifications on the combinatorial nature of representation are presented here;this is rela...With the challenge of quantum computing ahead, an analysis of number and representation adequate to the task is needed. Some clarifications on the combinatorial nature of representation are presented here;this is related to the foundations of digital representations of integers, and is thus also of interest in clarifying what numbers are and how they are used in pure and applied mathematics. The author hopes this work will help mathematicians and computer scientists better understand the nature of the Generalized Knapsack Code, a lattice-based code which the author believes to be particularly promising, and the use of number in computing in general.展开更多
Constant composition codes(CCCs)are a new generalization of binary constant weight codes and have attracted recent interest due to their numerous applications. In this paper, a new combinatorial approach to the constr...Constant composition codes(CCCs)are a new generalization of binary constant weight codes and have attracted recent interest due to their numerous applications. In this paper, a new combinatorial approach to the construction of CCCs is proposed, and used to establish new optimal CCCs.展开更多
Topology-transparent MAC scheduling strategies nowadays are all based on combinatorial design. To get throughput guarantee, a cover-free set is output as scheduling strategy of network. In this paper, we aim to modify...Topology-transparent MAC scheduling strategies nowadays are all based on combinatorial design. To get throughput guarantee, a cover-free set is output as scheduling strategy of network. In this paper, we aim to modify the cover-free set so that better throughput can be guaranteed. At the first step, the redundant slot of the cover-free set is proposed and found to have negative influence on the minimal guaranteed throughput. Second, we prove that any subset of a cover-free set is still a cover-free set after its redundant slots were squashed out. Our algorithm chooses the subset which has the maximal number of redundant slots, squashes all of its redundant slots, and then designates it as the network scheduling strategy. Therefore, better through- put can be guaranteed if the squashed subset is adopted as network scheduling strategy. For any topology- transparent node scheduling strategy, both the increased minimal throughput and decreased maximal transmission delay can be gotten by just using our algorithm as an extra accessory.展开更多
IT is known that the average undetected error probability （UEP） of a binary [n, k] code Con a binary symmetric channel with crossover probability p is given byP<sub>e</sub>（p） = sum from i=1 A<sub&g...IT is known that the average undetected error probability （UEP） of a binary [n, k] code Con a binary symmetric channel with crossover probability p is given byP<sub>e</sub>（p） = sum from i=1 A<sub>i</sub>p<sup>i</sup>（1-p）<sup>n-1</sup>, （1）where （A<sub>0</sub>, A<sub>1</sub>,…, A<sub>n</sub>） is the weight distribution of C. If for all 0≤p≤0.5, P<sub>e</sub>（p）≤P<sub>e</sub>（0. 5） = 2<sup>k-n</sup>-2<sup>-n</sup>, then C is called good for error detection. Moreover, if P<sub>e</sub> （p） ismonotonously increasing in the interval [0, 0.5], then C is called proper. Clearly, propercodes are good.展开更多
Two authentication codes with arbitration (A 2 codes) are constructed from finite affine spaces to illustrate for the first time that the information theoretic lower bounds for A 2 codes can be strictly tighter t...Two authentication codes with arbitration (A 2 codes) are constructed from finite affine spaces to illustrate for the first time that the information theoretic lower bounds for A 2 codes can be strictly tighter than the combinatorial ones. The codes also illustrate that the conditional combinatorial lower bounds on numbers of encodingdecoding rules are not genuine ones. As an analogue of 3 dimensional case, an A 2 code from 4 dimensional finite projective spaces is constructed, which meets both the information theoretic and combinatorial lower bounds.展开更多
Optical orthogonal code (OOC) has good correlation properties. It has many important appli-cations in a fiber-optic code-division multiple access channel. In this paper, a combinatorial construction foroptimal (15p, 5...Optical orthogonal code (OOC) has good correlation properties. It has many important appli-cations in a fiber-optic code-division multiple access channel. In this paper, a combinatorial construction foroptimal (15p, 5, 1) optical orthogonal codes with p congruent to 1 modulo 4 and greater than 5 is given byapplying Weil's Theorem. From this, when v is a product of primes congruent to 1 modulo 4 and greater than5, an optimal (15v, 5, 1)-OOC can be obtained by applying a known recursive construction.展开更多
文摘With the challenge of quantum computing ahead, an analysis of number and representation adequate to the task is needed. Some clarifications on the combinatorial nature of representation are presented here;this is related to the foundations of digital representations of integers, and is thus also of interest in clarifying what numbers are and how they are used in pure and applied mathematics. The author hopes this work will help mathematicians and computer scientists better understand the nature of the Generalized Knapsack Code, a lattice-based code which the author believes to be particularly promising, and the use of number in computing in general.
基金the National Natural Foundation of China (Grant Nos. 10671140, 10626038)
文摘Constant composition codes(CCCs)are a new generalization of binary constant weight codes and have attracted recent interest due to their numerous applications. In this paper, a new combinatorial approach to the construction of CCCs is proposed, and used to establish new optimal CCCs.
文摘Topology-transparent MAC scheduling strategies nowadays are all based on combinatorial design. To get throughput guarantee, a cover-free set is output as scheduling strategy of network. In this paper, we aim to modify the cover-free set so that better throughput can be guaranteed. At the first step, the redundant slot of the cover-free set is proposed and found to have negative influence on the minimal guaranteed throughput. Second, we prove that any subset of a cover-free set is still a cover-free set after its redundant slots were squashed out. Our algorithm chooses the subset which has the maximal number of redundant slots, squashes all of its redundant slots, and then designates it as the network scheduling strategy. Therefore, better through- put can be guaranteed if the squashed subset is adopted as network scheduling strategy. For any topology- transparent node scheduling strategy, both the increased minimal throughput and decreased maximal transmission delay can be gotten by just using our algorithm as an extra accessory.
文摘IT is known that the average undetected error probability （UEP） of a binary [n, k] code Con a binary symmetric channel with crossover probability p is given byP<sub>e</sub>（p） = sum from i=1 A<sub>i</sub>p<sup>i</sup>（1-p）<sup>n-1</sup>, （1）where （A<sub>0</sub>, A<sub>1</sub>,…, A<sub>n</sub>） is the weight distribution of C. If for all 0≤p≤0.5, P<sub>e</sub>（p）≤P<sub>e</sub>（0. 5） = 2<sup>k-n</sup>-2<sup>-n</sup>, then C is called good for error detection. Moreover, if P<sub>e</sub> （p） ismonotonously increasing in the interval [0, 0.5], then C is called proper. Clearly, propercodes are good.
文摘Two authentication codes with arbitration (A 2 codes) are constructed from finite affine spaces to illustrate for the first time that the information theoretic lower bounds for A 2 codes can be strictly tighter than the combinatorial ones. The codes also illustrate that the conditional combinatorial lower bounds on numbers of encodingdecoding rules are not genuine ones. As an analogue of 3 dimensional case, an A 2 code from 4 dimensional finite projective spaces is constructed, which meets both the information theoretic and combinatorial lower bounds.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10071056).
文摘Optical orthogonal code (OOC) has good correlation properties. It has many important appli-cations in a fiber-optic code-division multiple access channel. In this paper, a combinatorial construction foroptimal (15p, 5, 1) optical orthogonal codes with p congruent to 1 modulo 4 and greater than 5 is given byapplying Weil's Theorem. From this, when v is a product of primes congruent to 1 modulo 4 and greater than5, an optimal (15v, 5, 1)-OOC can be obtained by applying a known recursive construction.
