Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between ?-quasi-cyclic codes over a finite field ${\mathbb{F}}_q$ and linear codes over a ring $R={\mathbb{F}}_q[Y]/(Y^m?1)$. Using this correspondence, we prove that every ?-quasi-cyclic self-dual code of length m? over a finite field ${\mathbb{F}}_q$ can be obtained by the building-up construction, provided that $\mathrm{char}({\mathbb{F}}_q)=2$ or $q\equiv 1(\mathrm{mod}4)$, m is a prime p, and q is a primitive element of ${\mathbb{F}}_p$. We determine possible weight enumerators of a binary ?-quasi-cyclic self-dual code of length p? (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ?-quasi-cyclic codes of length 3?) optimal self-dual codes of lengths $30,36,42,48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40,20,12]$ code over ${\mathbb{F}}_3$ and a new 6-quasi-cyclic self-dual $[30,15,10]$ code over ${\mathbb{F}}_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28,14,9]$ code over ${\mathbb{F}}_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over ${\mathbb{F}}_4$.     

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$. These groups contain as special cases the odd dimensional general linear groups ${\mathrm{GL}}_{2n+1}(R)$ where R is any ring, the odd dimensional orthogonal and symplectic groups ${\mathrm{O}}_{2n+1}(R)$ and ${\mathrm{Sp}}_{2n+1}(R)$ where R is any commutative ring and further the first author's even dimensional unitary groups ${\mathrm{U}}_{2n}(R,\mathrm{\Lambda })$ where $(R,\mathrm{\Lambda })$ is any form ring. We classify the E-normal subgroups of the groups ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ (i.e. the subgroups which are normalized by the elementary subgroup ${\mathrm{EU}}_{2n+1}(R,\mathrm{\Delta })$), under the condition that R is either a semilocal or quasifinite ring with involution and $n\ge 3$. Further we investigate the action of ${\mathrm{U}}_{2n+1}(R,\mathrm{\Delta })$ by conjugation on the set of all E-normal subgroups.     

Non-level semi-standard graded Cohen–Macaulay domain with h-vector (h0,h1,h2)

Let $\mathbb{k}$ be an algebraically closed field of characteristic 0, and $A={?}_{i\in \mathbb{N}}{A}_i$ a Cohen–Macaulay graded domain with $A_0=\mathbb{k}$. If A is semi-standard graded (i.e., A is finitely generated as a $\mathbb{k}[A_1]$-module), it has the h-vector$(h_0,h_1,\dots ,h_s)$, which encodes the Hilbert function of A. From now on, assume that $s=2$. It is known that if A is standard graded (i.e., $A=\mathbb{k}[A_1]$), then A is level. We will show that, in the semi-standard case, if A is not level, then $h_1+1$ divides $h_2$. Conversely, for any positive integers h and n, there is a non-level A with the h-vector $(1,h,(h+1)n)$. Moreover, such examples can be constructed as Ehrhart rings (equivalently, normal toric rings).     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

Compact simple Lie groups admitting left-invariant Einstein metrics that are not geodesic orbit

In this article, we prove that the compact simple Lie groups $\mathrm{S}\mathrm{U}(n)$ for $n\ge 6$, $\mathrm{S}\mathrm{O}(n)$ for $n\ge 7$, $\mathrm{S}\mathrm{p}(n)$ for $n\ge 3$, ${\mathrm{E}}_6,{\mathrm{E}}_7,{\mathrm{E}}_8$, and ${\mathrm{F}}_4$ admit left-invariant Einstein metrics that are not geodesic orbit. This gives a positive answer to an open problem recently posed by Nikonorov.     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

On maximizing the fundamental frequency of the complement of an obstacle

Let $\mathrm{\Omega }?{\mathbb{R}}^n$ be a bounded domain satisfying a Hayman-type asymmetry condition, and let D be an arbitrary bounded domain referred to as an “obstacle”. We are interested in the behavior of the first Dirichlet eigenvalue ${\lambda }_1(\mathrm{\Omega }?(x+D))$.First, we prove an upper bound on ${\lambda }_1(\mathrm{\Omega }?(x+D))$ in terms of the distance of the set $x+D$ to the set of maximum points $x_0$ of the first Dirichlet ground state ${?}_{{\lambda }_1}>0$ of Ω. In short, a direct corollary is that if

(1)${\mu }_{\mathrm{\Omega }}:=\underset{x}{\max }?{\lambda }_1(\mathrm{\Omega }?(x+D))$

is large enough in terms of ${\lambda }_1(\mathrm{\Omega })$, then all maximizer sets $x+D$ of ${\mu }_{\mathrm{\Omega }}$ are close to each maximum point $x_0$ of ${?}_{{\lambda }_1}$.Second, we discuss the distribution of ${?}_{{\lambda }_1(\mathrm{\Omega })}$ and the possibility to inscribe wavelength balls at a given point in Ω.Finally, we specify our observations to convex obstacles D and show that if ${\mu }_{\mathrm{\Omega }}$ is sufficiently large with respect to ${\lambda }_1(\mathrm{\Omega })$, then all maximizers $x+D$ of ${\mu }_{\mathrm{\Omega }}$ contain all maximum points $x_0$ of ${?}_{{\lambda }_1(\mathrm{\Omega })}$.     

Scaling and non-standard matching theorems

Consider the standard Gaussian measure μ on ${\mathbb{R}}^2$. Consider independent r.v.s ${(X_i)}_{i\le N}$ distributed according to μ, and an independent copy ${(Y_i)}_{i\le N}$ of these r.v.s. We prove that, for some number C and N large, we have

(1)$\frac{{(\log ?N)}^2}{C}\le \mathsf{E}\underset{\pi }{\inf }?\sum _{i\le N}d{(X_i,Y_{\pi (i)})}^2\le C{(\log ?N)}^2,$

where the infimum is over all permutations π of $\{1,\dots ,N\}$. The striking point of this result is the factor ${(\log ?N)}^2$. Indeed, if instead of μ we consider the uniform distribution on the unit square, it is well known that the proper factor is $\log ?N$. The upper bound was proved by Michel Ledoux (2017) [3].