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.

We study the circulant complex Hadamard matrices of order n whose entries are l-th roots of unity. For n = l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n = p+ q, l = pq with p, q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n, l.

Wendt's binomial circulant determinant, W„ , is the determinant of an m by m circulant matrix of integers, with {i, ;')th entry (i,TM.i) whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m < 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = m...

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...

D-optimal designs are n x n ±l-matrices where n == 2 mod 4 with maximum determinant. D-optimal designs obtained via circulant matrices are equivalent to 2-{ v; kl i k2 i k1 + k2 ~(v 1)} supplementary difference sets, where v = ~. We use cyclotomy to construct D-optimal designs, where v is a prime. We give a generalisation of cyclotomy and extend the cyclotomic techniques which enables use to fi...

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices An. The n 2 th column of our circulant preconditioner Sn is equal to the n 2 th column of the given matrix An. Thus if An is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be deened as (S n Sn)...

Let n be any fixed positive integer. Every circulant weighing matrix of weight n arises from what we call an irreducible orthogonal family of weight n. We show that the number of irreducible orthogonal families of weight n is finite and thus obtain a finite algorithm for classifying all circulant weighing matrices of weight n. We also show that, for every odd prime power q, there are at most fi...

Abstract: Let be a Gaussian Fibonacci skew-circulant matrix, and be a Gaussian Fibonacci left skew-circulant matrix, and both of the first rows are , where is the th Gaussian Fibonacci number, and is a nonnegative integer. In this paper, by constructing the transformation matrices, the explicit determinants of and are expressed. Moreover, we discuss the singularities of these matrices and the i...

It is well-known that for each prime power q and for each d ∈ 2N, there exists a circulant weighing matrix of order q d+1−1 q−1 and weight q . We extend this result to show that there exist φ(d+1) 2 inequivalent circulant weighing matrices of order q d+1−1 q−1 and weight q , where φ is the Euler totient function. Further, we obtain a bound on the magnitude of the values taken by the cross-corre...