In this paper, we investigate the reduced form of circulant matrices and we show that the problem of computing the q-th roots of a nonsingular circulant matrix A can be reduced to that of computing the q-th roots of two half size matrices B - C and B + C.

In this paper we present a method to search q circulant matrices; the concatenation of these circulant matrices with circulant identity matrix generates quasi-cyclic codes with high various code rate q/(q+1) (q an integer). This method searches circulant matrices in order to find the good quasi-cyclic code (QCC) having the largest minimum distance. A modified simulated annealing algorithm is us...

In this paper we consider numerical solution of 3D linear elasticity equations described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) dis...

In this paper we consider numerical solution of 3D linear elasticity equations described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) dis...

This paper proposes a new, large diffusion layer for the AES block cipher. This new layer replaces the ShiftRows and MixColumns operations by a new involutory matrix in every round. The objective is to provide complete diffusion in a single round, thus sharply improving the overall cipher security. Moreover, the new matrix elements have low Hamming-weight in order to provide equally good perfor...

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...

In this paper we rst review the basic computational properties of per-manents, and then address some problems concerning permanents of (0; 1) circulant matrices. In particular we analyze their role at the boundary between computational tractability and intractability, showing that (i) a generic circulant matrix contains large arbitrary submatrices, a fact which casts some doubt on the tractabil...

In this paper, we propose a new low-rank tensor model based on the circulant algebra, namely, twist tensor nuclear norm or t-TNN for short. The twist tensor denotes a 3-way tensor representation to laterally store 2D data slices in order. On one hand, t-TNN convexly relaxes the tensor multi-rank of the twist tensor in the Fourier domain, which allows an efficient computation using FFT. On the o...

For a given nonnegative integer g , a matrix An of size n is called g-Toeplitz if its entries obey the rule An =  ar−gs n−1 r,s=0. Analogously, a matrix An again of size n is called g-circulant if An =  a(r−gs) mod n n−1 r,s=0. In a recent work we studied the asymptotic properties, in terms of spectral distribution, of both g-circulant and g-Toeplitz sequences in the case where {ak} can be ...