Copyright (c) 2009 All rights reserved. http://works.bepress.com/jseberry Recent documents in Prof. Jennifer R. Seberry en-us Sun, 31 May 2009 07:49:08 PDT 3600 http://works.bepress.com/jseberry/112 http://works.bepress.com/jseberry/112 Wed, 28 May 2008 16:31:38 PDT We give new sets of sequences with entries from {0, ±a, ±b, ±c} on the commuting variables a, b, c and zero autocorrelation function. Then we use these sequences to construct some new orthogonal de-signs. We show the necessary conditions for the existence of an OD(28; s1, s2, s3) plus the condition that (s1, s2, s3) ≠ (1,5,20) are sufficient conditions for the existence of an OD(28; s1, s2, s3). We also show the necessary conditions for the existence of an OD(28; s1, s2, s3) constructed using four circulant matrices are sufficient conditions for the existence of 4 -- NPAF(s1, s2, s3) sequences of of length n for all lengths n ≥ 7. We establish asymptotic existence results for OD(4N; s1, s2) for 2 ≤ s1 + s2 ≤ 28. We show the necessary conditions for the existence of an OD(28; s1, s2) with 25 ≤ s1 + s2 ≤ 28, constructed using four circulant matrices, plus the condition that (s1, s2) ≠ (1,26), (2, 25), (7, 19), (8, 19) or (13, 14), are sufficient conditions for the existence of 4 -- NPAF(s1, s2) sequences of of length n for all lengths n ≥ 7. C. Koukouvinos http://works.bepress.com/jseberry/111 http://works.bepress.com/jseberry/111 Wed, 28 May 2008 16:31:34 PDT We prove that cubic homogeneous bent functions f : V2n → GF(2) exist for all n ≥ 3 except for n = 4. J. Seberry http://works.bepress.com/jseberry/110 http://works.bepress.com/jseberry/110 Wed, 28 May 2008 16:31:29 PDT H. Kharaghani, in "Arrays for orthogonal designs", J. Combin. Designs, 8 (2000), 166-173, showed how to use amicable sets of matrices to construct orthogonal designs in orders divisible by eight. We show how amicable orthogonal designs can be used to make amicable sets and so obtain infinite families of orthogonal designs in six variables in orders divisible by eight. C. Koukouvinos http://works.bepress.com/jseberry/109 http://works.bepress.com/jseberry/109 Wed, 28 May 2008 16:31:25 PDT We use a new algorithm to find new sets of sequences with entries from {0, ±a, ±b, ±c, ±d}, on the commuting variables a, b, c, d, with zero autocorrelation function. Then we use these sequences to construct a series of new three and four variable orthogonal designs in order 36. We show that the necessary conditions plus (.s1, s2, s3, s4) not equal to 12816 18 816 221313 26721 36 816 4889 12825 191313 23 424 289 9 381015 8899 14425 22 916 are sufficient for the existence of an OD(36; s1, s2 s3, s4) constructed using four circulant matrices in the Goethals-Seidel array. Of the 154 theoretically possible cases 133 are known. We also show that the necessary conditions plus (s1, s2 s3) ≠ (2,8,25), (6,7,21), (8, 9, 17) or (9, 13, 13) are sufficient for the existence of an OD(36; s1, s2, s3) constructed using four circulant matrices in the Goethals-Seidel array. Of the 433 theoretically possible cases 429 are known. Further, we show that the necessary conditions are sufficient for the existence of an OD(36; s1, s2 36 -- s1 -- s2) in each of the 54 theoretically possible cases. Further, of the 27 theoretically possible OD(36; s1, s2, s3, 36 -- s1 -- s2 -- s3), 23 are known to exist, and four, (1,2,8,25), (1, 9,13,13), (2,6,7,21) and (3, 8,10,15), cannot be constructed using four circulant matrices. By suitably replacing the variables by ±1 these lead to more than 200 potentially inequivalent Hadamard matrices of order 36. By considering the 12 OD(36; 1, s1, 35--s1) and suitably replacing the variables by ±1 we obtain 48 potentially inequivalent skew-Hadamard matrices of order 36. A summary with all known results in order 36 is presented in the Tables. S. Georgiou http://works.bepress.com/jseberry/108 http://works.bepress.com/jseberry/108 Wed, 28 May 2008 16:31:20 PDT Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Grobner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of computing a kind of a particular basis of an ideal which enjoys some important properties. In this paper we introduce the concept of Hadamard ideals in order to establish a new approach to the construction of Hadamard matrices with circulant core. Hadamard ideals reveal the rich interplay between Hadamard matrices with circulant core and ideals in multivariate polynomial rings. Our approach yields an exhaustive list of Hadamard matrices with circulant core for any specific dimension. I. S. Kotsireas http://works.bepress.com/jseberry/107 http://works.bepress.com/jseberry/107 Wed, 28 May 2008 16:31:16 PDT In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n -- j) x (n -- j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given. C. Koukouvinos http://works.bepress.com/jseberry/106 http://works.bepress.com/jseberry/106 Wed, 28 May 2008 16:31:12 PDT Recently K. T. Arasu (personal communication) and Yoseph Strassler, in his PhD thesis, The Classification of Circulant Weighing Matrices of Weight 9, Bar-Ilan University, Ramat-Gan, 1997, have intensively studied circulant weighing matrices, or single sequences, with weight 9. They show many cases are non-existent. Here we give details of a search for two sequences with zero periodic autocorrelation and types (1,9), (1,16) and (4,9). We find some new cases but also many cases where the known necessary conditions are not sufficient. We instance a number of occasions when the known necessary conditions are not sufficient for the existence of weighing matrices and orthogonal de-signs constructed using sequences with zero autocorrelation function leading to intriguing new questions. J. Horton http://works.bepress.com/jseberry/105 http://works.bepress.com/jseberry/105 Wed, 28 May 2008 16:31:08 PDT Some new amicable orthogonal designs of order 8 are found as part of a complete search of the equivalence classes for orthogonal designs OD(8; 1,1,1,1), OD(8; 1,1,1,4), OD(8; 1,1,2,2), OD(8; 1,1,1,2), OD (8; 1,1,2,4), OD(8; 1,1,1,3), OD(8; 1,1,2,3), OD(8; 1,1,1,5) and OD(8; 1,1,3,3). Y. Zhao http://works.bepress.com/jseberry/104 http://works.bepress.com/jseberry/104 Wed, 28 May 2008 16:31:03 PDT We establish that the necessary conditions for the existence of Bhaskar Rao designs of block size five are : i). λ(v - 1) ≡ 0 (mod 4) ii). λv(v - 1) ≡ 0 (mod 40) iii). 2|λ. We show these conditions are sufficient: for λ = 4 if v > 215, with 10 smaller possible exceptions and one definite exception at v = 5; for λ = 10 if v > 445, with 11 smaller possible exceptions, and one definite exception at v = 5; and for λ = 20, with the possible exception of v = 32; we also give a few results for other values of λ. G. R. Chaudhry http://works.bepress.com/jseberry/103 http://works.bepress.com/jseberry/103 Wed, 28 May 2008 16:30:59 PDT In a large system, XML documents associated with it can be large and complicated. To manage access control in such a large and complicated system is very difficult. Recently, Access Policy Sheet (APS) [6] was introduced to provide a solution to access control for large XML systems. In this paper, we proposed a temporal access control scheme in APS where the propagation of authorization rights is assumed. The authorization policies can be automatically revoked when the associated time expires. We also provide conflict resolutions for our temporal authorization system. J. Wu