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...

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.

It is known that if an almost bipartite graph G with n edges possesses a γlabeling, then the complete graphK2nx+1 admits a cyclicG-decomposition. We introduce a variation of γ-labeling and show that whenever an almost bipartite graph G admits such a labeling, then there exists a cyclic Gdecomposition of a family of circulant graphs. We also determine which odd length cycles admit the variant la...

It is known that if a bipartite graph G with n edges possesses any of three types of ordered labelings, then the complete graphK2nx+1 admits a cyclic G-decomposition for every positive integer x. We introduce variations of the ordered labelings and show that whenever a bipartite graph G admits one of these labelings, then there exists a cyclic G-decomposition of an infinite family of circulant ...

We estimate the norms of standard Gaussian random Toeplitz and circulant matrices and their inverses, mostly by means of combining some basic techniques of linear algebra. In the case of circulant matrices we obtain sharp probabilistic estimates, which show that these matrices are expected to be very well conditioned. Our probabilistic estimates for the norms of standard Gaussian random Toeplit...

Circulant matrices play a central role in a recently proposed formulation of three-way data computations. In this setting, a three-way table corresponds to a matrix where each “scalar” is a vector of parameters defining a circulant. This interpretation provides many generalizations of results from matrix or vector-space algebra. These results and algorithms are closely related to standard decou...