This note reports work in progress in connection with Ryser’s conjecture on circulant Hadamard matrices.

The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant m...

Locke and Witte in 9] have described a class of nonhamiltonian circulant digraphs. We show that for innnitely many of them the reversal of any arc produces a hamil-tonian cycle. This solves an open problem stated in 4]. We use these graphs to construct counterexamples to Add am's conjecture. The smallest one Cay(Z 12 ; 2; 3; 8) 4 is the counterexample with the smallest known number of vertices.

In order to construct quantum [[n, 0, d]] codes for (n, d) = (56, 15), (57, 15), (58, 16), (63, 16), (67, 17), (70, 18), (71, 18), (79, 19), (83, 20), (87, 20), (89, 21), (95, 20), we construct self-dual additive F4-codes of length n and minimum weight d from circulant graphs. The quantum codes with these parameters are constructed for the first time.

For a given graph G, denote by A its adjacency matrix and F (t) = exp(iAt). We say that there exists a perfect state transfer (PST) in G if |F (τ)ab| = 1, for some vertices a, b and a positive real number τ . Such a property is very important for the modeling of quantum spin networks with nearest-neighbor couplings. We consider the existence of the perfect state transfer in integral circulant g...

Circulant graphs are an important class of interconnection networks in parallel and distributed computing. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer as well. The integral circulant graph ICGn(D) has the vertex set Zn = {0, 1, 2, . . . , n− 1} and vertices a and b are adjacent if gcd(a− b, n) ∈ D, where D ⊆ {d : d | n...

The Circulant Embedding Method (CEM) is a well known technique to generate stationary Gaussian Random Fields (GRF). main idea embed the covariance matrix in larger nested block circulant matrix, whose factorization can be rapidly computed thanks fast Fourier transform (FFT) algorithm. CEM requires extended at least positive semidefinite which proven case if enclosing domain sufficiently large, ...

Algebraic techniques are employed to obtain necessary conditions for the existence of certain circulant weighing matrices. As an application we rule out the existence of many circulant weighing matrices. We study orders n = 8 +8+1, for 10 ~ 8 ~ 25. These orders correspond to the number of points in a projective plane of order 8.

We give a classification of all double circulant doubly even self-dual [72, 36, 12] codes. We also demonstrate that every double circulant doubly even self-dual [72, 36, 12] code has no extremal doubly even self-dual [72, 36, 16] neighbor.