In this paper we completely classify the circulant weighing matrices of weight 16 and odd order. It turns out that the order must be an odd multiple of either 21 or 31. Up to equivalence, there are two distinct matrices in CW(31,16), one matrix in CW(21,16) and another one in CW(63,16) (not obtainable by Kronecker product from CW(21,16)). The classification uses a multiplier existence theorem.
A circulant weighing matrix $W = (w_{i,j})$ is a square matrix of order $n$ and entries $w_{i,j}$ in $\{0, \pm 1\}$ such that $WW^T=kI_n$. In his thesis, Strassler gave a table of existence results for such matrices with $n \leq 200$ and $k \leq 100$. In the latest version of Strassler's table given by Tan \cite{arXiv:1610.01914} there are 34 open cases remaining. In this paper we give nonexistence proofs for 12 of these cases, report on preliminary searches outside Strassler's table, and characterize the known proper circulant weighing matrices.
Gauss periods taking exactly two values are closely related to two-weight irreducible cyclic codes and strongly regular Cayley graphs. They have been extensively studied in the work of Schmidt and White and others. In this paper, we consider the question of when Gauss periods take exactly three rational values. We obtain numerical necessary conditions for Gauss periods to take exactly three rational values. We show that in certain cases, the necessary conditions obtained are also sufficient. We give numerous examples where the Gauss periods take exactly three vlaues. Furthermore, we discuss co…
A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors ($k$-coloring) is a map from its vertex set to the set of integers $\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. We consider perfect colorings of Cayley graphs of the additive group of integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a positive integer $n$. We enumerate perfect $2$-colorings of the graphs under consideration and state the conjecture generalizing the main result to an arbitrary…
Thin divisible designs graphs: an interplay between fixed-point free involutions of $(v,k,λ)$-graphs and symmetric weighing matrices
Hebei Normal University（河北师范大学）；Tilburg University（蒂尔堡大学）；Sobolev Institute of Mathematics（索博列夫数学研究所）；China Three Gorges University（三峡大学）；Novosibirsk State University（新西伯利亚国立大学）；Jiangxi Normal University（江西师范大学）
Circulant matrices with orthogonal rows and off-diagonal entries of absolute value $1$
已知具有对角元素$d\geq0$、非对角元素$\pm1$和正交行的实对称循环矩阵仅存在$2d+2$阶（以及$1$阶）[Turek and Goyeneche 2019]。在本文中，我们考虑了一个复杂的埃尔米特类似的这些矩阵。也就是说，我们研究了具有正交行，对角元$d\geq0$和任何绝对值$1$不在对角元上的复元的厄米特循环矩阵的存在性和构造。作为一个特殊的情况下，我们考虑矩阵的非对角元素是4次根的单位，我们证明了任何这样的矩阵与$d$不同于奇数的顺序是$n=2d+2$。我们还讨论了有限环$\mathbb{Z}_m$上对称循环矩阵的一个类似问题。作为我们的结果的应用，我们显示了一个密切的联系，相互无偏基地，在量子信息论的一个重要的开放问题。
Universitat de Lleida（莱里达大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；Barcelona Graduate School of Mathematics（巴塞罗那数学研究生院）；Institut de Matemàtiques de la UPC-BarcelonaTech (IMTech)（UPC-巴塞罗那理工大学数学研究所）
It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and an even unicyclic graph are quite different, we consider wheel graphs as they are formed from odd or even cycles. In this article solve the open problem for wheel graphs. This work has an interesting connection to inverses of circulant matrices.
A weighing matrix $W$ of order $n=\frac{p^{m+1}-1}{p-1}$ and weight $p^m$ is constructed and shown that the rows of $W$ and $-W$ form optimal constant weight ternary codes of length $n$, weight $p^m$ and minimum distance $p^{m-1}(\frac{p+3}{2})$ for each odd prime power $p$ and integer $m\ge 1$ and thus $$A_3\left(\frac{p^{m+1}-1}{p-1},p^{m-1}\big(\frac{p+3}{2}\big),p^{m}\right)=2\big(\frac{p^{m+1}-1}{p-1}\big).$$
