New Self-Dual Codes from 2x2 Block Circulant Matrices, Group Rings and Neighbours of Neighbours

A closed-form expression for the kth power of semicirculant and r-circulant matrices

We derive a closed-form expression for the kth power of semicirculant matrices by using the determinant of certain matrices. As an application, a closed-form expression for the kth power of r-circulant matrices is also povided.

Vector-Circulant Matrices over Finite Fields and Related Codes

A vector-circulant matrix is a natural generalization of the classical circulant matrix and has applications in constructing additive codes. This article formulates the concept of a vector-circulant matrix over finite fields and gives an algebraic characterization for this kind of matrix. Finally, a construction of additive codes with vector-circulant based over $\mathbb{F}_4$ is given together with some examples of good half-rate additive codes.

Arbitrary positive powers of semicirculant and r-circulant matrices

We provide a novel recursive method, which does not require any assumption, to compute the entries of the kth power of a semicirculant matrix. As an application, a method for computing the entries of the kth power of r-circulant matrices is also presented.

Determinants of Binomial-Related Circulant Matrices

Due to their rich algebraic structures and various applications, circulant matrices have been of interest and continuously studied. In this paper, the notions of Binomial-related matrices have been introduced. Such matrices are circulant matrices whose the first row is the coefficients of $(x+zy)^n$, where $z$ is a complex number of norm $1$ and $n$ is a positive integer. In the case where $z\in \{1,-1,i,-i\}$, the explicit formula for the determinant of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as special cases. Fin…

Reachability of eigenspaces for interval circulant matrices in max-algebra

A nonnegative matrix A is said to be strongly robust if its max-algebraic eigencone is universally reachable, i.e., if the orbit of any initial vector ends up with a max-algebraic eigenvector of A. Consider the case when the initial vector is restricted to an interval and A can be any matrix from a given interval of nonnegative circulant matrices. The main aim of this paper is to classify and characterize the six types of interval robustness in this situation. This naturally leads us also to study the max-algebraic spectral theory of circulant matrices and the relation of inclusion between att…

Critical groups of generalized de Bruijn and Kautz graphs and circulant matrices over finite fields: an extended abstract

We determine the critical groups of the generalized de Bruijn graphs DB$(n,d)$ and generalized Kautz graphs Kautz$(n,d)$, thus extending and completing earlier results for the classical de Bruijn and Kautz graphs. Moreover, for a prime $p$ the critical groups of DB$(n,p)$ are shown to be in close correspondence with groups of $n\times n$ circulant matrices over $\mathbb{F}_p$, which explains numerical data in [OEIS:A027362], and suggests the possibility to construct normal bases in $\mathbb{F}_{p^n}$ from spanning trees in DB$(n,p)$.

$G$-循环矩阵与经典 Maschke 定理

$G$-circulant Matrices and the Classical Maschke Theorem

该研究借助域上 $G$-循环矩阵环与群环 $k[G]$ 的同构关系，为经典 Maschke 定理提供了一个简洁证明。

Determinants of some Special Matrices over Commutative Finite Chain Rings

Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings $R$ with residue field $\mathbb{F}_q$ are studied. The number of $n\times n$ diagonal matrices over ${R}$ of determinant $a$ is determined for all elements $a$ in $ {R}$ and for all positive integers $n$. Subsequently, …

Approximating real symmetric Toeplitz matrices using the nearest circulant

The nearest circulant approximation of a real Toeplitz matrix in the Frobenius norm is derived. This matrix is symmetric. It is proven that symmetric circulant matrices are the only real circulant matrices with all real eigenvalues. The Frobenius norm of the difference between this approximation and the Toeplitz matrix for the case of a Toeplitz matrix displaying exponential decay is evaluated using an expression of $\sum_{n = 1}^N n^k p^n$ in terms of the first $k$ geometric moments. Compared to a classic approximation the nearest circulant displays dramatically better behaviour in any finite…

Block diagonalization of block circulant quaternion matrices and the fast calculation for T-product of quaternion tensors

With the great success of the T-product based real tensor methods in the color image and gray video processing, the establishment of T-product based quaternion tensor methods in the color video processing has encountered a challenge, which is the block diagonalization of block circulant quaternion matrices. In this paper, we show that the discrete Fourier matrix $\mathbf{F_p}$ cannot diagonalize $p\times p$ circulant quaternion matrices, nor can the unitary quaternion matrices $\mathbf{F_p}\mathbf{j}$ and $\mathbf{F_p}(1+\mathbf{j})/\sqrt{2}$ with $\mathbf{j}$ being an imaginary unit of quater…

Generalized hyperbolic functions, circulant matrices and functional equations

There is a contrast between the two sets of functional equations f_0(x+y) = f_0(x)f_0(y) + f_1(x)f_1(y), f_1(x+y) = f_1(x)f_0(y) + f_0(x)f_1(y), and f_0(x-y) = f_0(x)f_0(y) - f_1(x)f_1(y), f_1(x-y) = f_1(x)f_0(y) - f_0(x)f_1(y) satisfied by the even and odd components of a solution of f(x+y) = f(x) f(y). J. Schwaiger and, later, W. Förg-Rob and J. Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.

A Circulant Approach to Skew-Constacyclic Codes

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x;θ] modulo the left ideal generated by a polynomial of the type x^n-a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This recovers the well-known result that the dual of a skew-constacyclic code is a constacyclic code again. Special attention is paid to the case where x^n-a is two-sided.