We present a square root algorithm in Fq which generalizes Atkins’s square root algorithm [6] for q ≡ 5 (mod 8) and Kong et al.’s algorithm [8] for q ≡ 9 (mod 16). Our algorithm precomputes a primitive 2-th root of unity ξ where s is the largest positive integer satisfying 2|q − 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square roo...

We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here “binary matrix” means a matrix whose elements are drawn from {0, 1} or {−1, 1}. We describe efficient parallel algorithms for the search, using Duval’s algorithm for generation of Lyndon words and the well-known representation of the determinant of a circulant in terms of roots of unity....

In this paper we present a time-polynomial recognition algorithm for circulant graphs of prime order. A circulant graphG of order n is a Cayley graph over the cyclic groupZn: Equivalently, G is circulant i its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent con guration A and, in particular, a S...

Circulant matrices with general shift by k places have been studied in the literature and formula for their eigenvalues is known. We first reestablish this formula and some further properties of these eigenvalues in a manner suitable for our use. We then consider random k = k(n) circulants Ak,n with n → ∞ and whose input sequence {ai} is independent with mean zero and variance one and supn n −1...

In this paper, we discuss the solutions to a class of Hermitian positive deenite system Ax = b by the preconditioned conjugate gradient method with circulant preconditioner C. In general, the smaller the condition number (C ?1=2 AC ?1=2) is, the faster the convergence of the method will be. The circulant matrix C b that minimizes (C ?1=2 AC ?1=2) is called the best conditioned circulant precond...

We present a new construction for two-dimensional, perfect autocorrelation arrays over roots of unity. These perfect arrays are constructed from a block of perfect column sequences. Other blocks are constructed from the first block, to generate a block-circulant structure. The columns are then multiplied by a perfect sequence over roots of unity, which, when folded into an array commensurate wi...

1 Abstract In this paper we present a time-polynomial recognition algorithm for circulant graphs of prime order. A circulant graph G of order n is a Cayley graph over the cyclic group Z n : Equivalently, G is circulant ii its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent connguration A and, in...

We show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finiteand infinite-dimensional maximum entropy problems concerning probability distributions, spectral densities, and covariance matrices. These include Burg’s spectral estimation method and Dempster’s covariance completion, as well as various recent generalizations of the above. We th...