In this article, we analyze the circulant structure of generalized circulant matrices to reduce the search space for finding lightweight MDS matrices. We first show that the implementation of circulant matrices can be serialized and can achieve similar area requirement and clock cycle performance as a serial-based implementation. By proving many new properties and equivalence classes for circul...

We prove that there is no circulant Hermitian complex Hadamard matrix of order n > 4.

A framework for constructing circulant and block circulant preconditioners (C) for a symmetric linear system Ax= b arising from signal and image processing applications is presented in this paper. The proposed scheme does not make explicit use of matrix elements of A. It is ideal for applications in which A only exists in the form of a matrix vector multiplication routine, and in which the proc...

We consider the number of spanning trees in circulant graphs of βn vertices with generators depending linearly on n. The matrix tree theorem gives a closed formula of βn factors; while we derive a formula of β−1 factors. The spanning tree entropy of these graphs is then compared to the one of fixed generated circulant graphs.

In the second part of this paper we study condition numbers with respect to componentwise perturbations in the input data for linear systems and for matrix inversion, and the distance to the nearest singular matrix. The structures under investigation are linear structures, namely symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Ha...

In present paper, we investigate 4 problems. Firstly, it is known that, a matrix is MDS if and only if all sub-matrices of this matrix of degree from 1 to n are full rank. In this paper, we propose a theorem that an orthogonal matrix is MDS if and only if all sub-matrices of this orthogonal matrix of degree from 1 to bn2 c are full rank. With this theorem, calculation of constructing orthogonal...

We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the matrix a PGD can have at most three distinct eigenvalues, all which are nonnegative integers. The contains useful information on incidence structure design. An ordinary 2- ( v , k λ ) $(v,k,\lambda )$ design has single $\lambda $ and its circulant, geometry two...

It is known that extremal doubly-even self-dual codes of length n ≡ 8 or 0 (mod 24) yield 3or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) ∈ {(32, 8, 5), (56, 12, 9), (56, 16, 9), (56, 24, 9), (80, 16, 52)}. With the aid of computer, we obtain 22 generator matrices of bor...

We introduce the class of skew-circulant lattice rules. These are s-dimensional lattice rules that may be generated by the rows of an s×s skewcirculant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d. We describe so...

We describe the results of a computer-based search for 5 and 6dimensional lattice rules of specified trigonometric degree. In this search only lattice rules that can be generated by a circulant or skew-circulant generator matrix are considered, which makes this approach significantly faster than earlier approaches. The drawback is that we do not necessarily obtain optimal lattice rules. We also...