The use of amicable sets of eight circulant matrices and four negacyclic matrices to generate orthogonal designs is relatively new. We find all 1841 possible orthogonal designs of order 40 in three variables, using only these new techniques.

Mobeen Munir 1, Waqas Nazeer 1, Zakia Shahzadi 1 and Shin Min Kang 2,3,* 1 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; [email protected] (M.M.); [email protected] (W.N.); [email protected] (Z.S.) 2 Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea 3 Center for General Educati...

A critical set in a latin square is a subset of its elements with the following properties: 1) No other latin square exists which also contains that subset. 2) No element may be deleted without destroying property 1. Let scs(n) denote the smallest possible cardinality of a critical set in an n × n latin square. It is conjectured that scs(n) = n/4 , and that only the back-circulant latin square ...

Cavenagh and Wanless [Discrete Appl. Math. 158 no. 2 (2010), 136–146] determined the possible intersection of any two transversals of the back circulant latin square Bn, and used the result to completely determine the spectrum for 2-way k-homogeneous latin trades. We generalize this problem to the intersection of μ transversals of Bn such that the transversals intersect stably (that is, the int...

This article identifies the general form of matrix power and trace an integer Special Form 3 × Complex Circulant matrix. The research begins to determine matrix, followed by determining proof is done using mathematical induction. final result this obtain A^n tr(A^n) for n integers in special complex

Long quasi-cyclic codes of any fixed index > 1 have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. Güneri, H. Shoaib, P. Solé, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly selfdual double circulant codes, and self-dual four circulant codes, have ...

We consider the problem of computing square root a perturbation scaled identity matrix, , where and are matrices with . This arises in various applications, including computer vision optimization methods for machine learning. derive new formula th that involves weighted sum powers matrix is particularly attractive root, since has just one term when also class Newton iterations exploit low-rank ...

Circulant matrices have attracted interest due to their rich algebraic structures and various applications. In this paper, the concept of vector-circulant matrices over finite fields is studied as a generalization of circulant matrices. The algebraic characterization for such matrices has been discussed. As applications, constructions of vector-circulant based additive codes over finite fields ...

Article history: Received 12 July 2007 Received in revised form 10 March 2008 Accepted 12 March 2008 Available online 18 March 2008 MSC: primary 62K15 secondary 62K05

We obtain several new number theoretic results which improve the field descent method. We use these results to rule out many of the known open cases of the circulant Hadamard matrix conjecture. In particular, the only known open case of the Barker sequence conjecture is settled.