Smallness of fundamental groups for arithmetic schemes

The smallness is proved of étale fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the Hermite-Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain finiteness results of representations of the fundamental groups over algebraically closed fields.     

Self-Orthogonal 3-(56,12,65) Designs and Extremal Doubly-Even Self-Dual Codes of Length 56

In this paper, we show that the code generated by the rows of a block-point incidence matrix of a self-orthogonal 3-(56,12,65) design is a doubly-even self-dual code of length 56. As a consequence, it is shown that an extremal doubly-even self-dual code of length 56 is generated by the codewords of minimum weight. We also demonstrate that there are more than one thousand inequivalent extremal doubly-even self-dual [56,28,12] codes. This result shows that there are more than one thousand non-isomorphic self-orthogonal 3-(56,12,65) designs. AMS Classification: 94B05, 05B05     

On the classification of weighing matrices and self‐orthogonal codes

We provide a classification method of weighing matrices based on a classification of self‐orthogonal codes. Using this method, we classify weighing matrices of orders up to 15 and order 17, by revising some known classification. In addition, we give a revised classification of weighing matrices of weight 5. A revised classification of ternary maximal self‐orthogonal codes of lengths 18 and 19 is also presented. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:40–57, 2012     

A Characterization of Designs Related to an Extremal Doubly-Even Self-Dual Code of Length 48

The uniqueness of a binary doubly-even self-dual [48, 24, 12] code is used to prove that a self-orthogonal 5-(48, 12, 8) design, as well as some of its derived and residual designs, including a quasi-symmetric 2-(45, 9, 8) design, are all unique up to isomorphism.Received November 5, 2003     

There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Self-dual codes over exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24, 12, 10] code over , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.      

On first passage time structure of random walks

For continuous time birth-death processes on {0,1,2,…}, the first passage time T⁺_n from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T_0,n+1 from 0 to (n + 1) is always a sum of (n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T⁺_n and T_0,n+1 for discrete time birth-death chains become a mixture and a sum, respectively, of (n + 1) independent geometric random variables on {1,2,…};. The results are further extended to conditional first passage times.     

Extremal self-dual codes of length 64 through neighbors and covering radii

We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.     

New Optimal Self-Dual Codes over GF(7)

In this paper, double circulant self-dual codes over GF(7) are presented, including [12,6,6] codes which are new optimal codes. It is shown that the supports of the codewords of weights 9, 10 and 11 in double circulant [20,10,9] codes form 3-designs. For larger lengths, some good self-dual codes are constructed from weighing matrices. Received: June 24, 1996 / Revised: February 28, 1997     

Analysis of effective service time with age dependent interruptions and its application to optimal rollback policy for database management

A stochastic model is developed describing a service system subject to inhomogeneous Poisson interruptions with age dependent interruption periods. By studying the probabilistic flow of the underlying multivariate Markov process, the Laplace transform of the effective service time is explicitly obtained. For general renewal interruptions, only the expected effective service time is derived. As an application, an optimal checkpoint policy is examined for database management. It is shown that an optimal policy maximizing the ergodic availability of the database is to implement a checkpoint as soon as the cumulative uptime of the database reaches a prespecified constantk ^*. A computational procedure is then developed for findingk ^* and numerical results are exhibited.This work was supported in part by the National Science Foundation under Grant No. ECS-8600992 and by the IBM Program of Support for Education in the Management of Information Systems.     

Mutually disjoint designs and new 5‐designs derived from groups and codes

The article gives constructions of disjoint 5‐designs obtained from permutation groups and extremal self‐dual codes. Several new simple 5‐designs are found with parameters that were left open in the table of 5‐designs given in (G. B. Khosrovshahi and R. Laue, t‐Designs with t⩾3, in “Handbook of Combinatorial Designs”, 2nd edn, C. J. Colbourn and J. H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 79–101), namely, 5−(v, k, λ) designs with (v, k, λ)=(18, 8, 2m) (m=6, 9), (19, 9, 7m) (m=6, 9), (24, 9, 6m) (m=3, 4, 5), (25, 9, 30), (25, 10, 24m) (m=4, 5), (26, 10, 126), (30, 12, 440), (32, 6, 3m) (m=2, 3, 4), (33, 7, 84), and (36, 12, 45n) for 2⩽n⩽17. These results imply that a simple 5−(v, k, λ) design with (v, k)=(24, 9), (25, 9), (26, 10), (32, 6), or (33, 7) exists for all admissible values of λ. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 305–317, 2010