Maximum distance separable codes over {\mathbb{Z}_4} and {\mathbb{Z}_2 \times \mathbb{Z}_4}

Known upper bounds on the minimum distance of codes over rings are applied to the case of ${\mathbb Z_{2}\mathbb Z_{4}}$ -additive codes, that is subgroups of ${\mathbb Z_{2}^{\alpha}\mathbb Z_{4}^{\beta}}$ . Two kinds of maximum distance separable codes are studied. We determine all possible parameters of these codes and characterize the codes in certain cases. The main results are also valid when ?? = 0, namely for quaternary linear codes.     

Selmer groups over {\mathbb {Z}}_p^d-extensions

Consider an abelian variety \(A\) defined over a global field \(K\) and let \(L/K\) be a \({\mathbb {Z}}_p^d\) -extension, unramified outside a finite set of places of \(K\) , with \({{\mathrm{Gal}}}(L/K)=\Gamma \) . Let \(\Lambda (\Gamma ):={\mathbb {Z}}_p[[\Gamma ]]\) denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the \(\Lambda (\Gamma )\) -module \(X_L\) , the dual \(p\) -primary Selmer group, varies when \(L/K\) is replaced by a strict intermediate \({\mathbb {Z}}_p^e\) -extension.     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

Structural adaptive deconvolution under $${\mathbb{L}_p}$$-losses

In this paper, we address the problem of estimating a multidimensional density f by using indirect observations from the statistical model Y = X + ε. Here, ε is a measurement error independent of the random vector X of interest and having a known density with respect to Lebesgue measure. Our aim is to obtain optimal accuracy of estimation under \({\mathbb{L}_p}\)-losses when the error ε has a characteristic function with a polynomial decay. To achieve this goal, we first construct a kernel estimator of f which is fully data driven. Then, we derive for it an oracle inequality under very mild assumptions on the characteristic function of the error ε. As a consequence, we getminimax adaptive upper bounds over a large scale of anisotropic Nikolskii classes and we prove that our estimator is asymptotically rate optimal when p ∈ [2,+∞]. Furthermore, our estimation procedure adapts automatically to the possible independence structure of f and this allows us to improve significantly the accuracy of estimation.     

Proving properties of matrices over {\mathbb{Z}_{2}}

We prove assorted properties of matrices over ${\mathbb{Z}_{2}}$ , and outline the complexity of the concepts required to prove these properties. The goal of this line of research is to establish the proof complexity of matrix algebra. It also presents a different approach to linear algebra: one that is formal, consisting in algebraic manipulations according to the axioms of a ring, rather than the traditional semantic approach via linear transformations.     

On the regular Sylow p-subgroups of Chevalley groups over \mathbb{Z}_{p^m }

We prove that a Sylow p-subgroup of the general linear group of dimension n over the residue ring modulo p ^m is regular for n ² < p and powerful if and only if n = 2 and m = 1. We obtain similar results for the Sylow p-subgroups of normal types over the same ring.     

Zero-temperature Glauber dynamics on {\mathbb{Z}^d}

We study zero-temperature Glauber dynamics on ${\mathbb{Z}^d}$ , which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define ${p_c(\mathbb{Z}^d)}$ to be the infimum over p such that the system fixates at ???+??? with probability 1. It is a folklore conjecture that ${p_c(\mathbb{Z}^d) = 1/2}$ for every ${2 \le d \in \mathbb{N}}$ . We prove that ${p_c(\mathbb{Z}^d) \to 1/2}$ as d ?? ??.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

A Construction of Partial Difference Sets in {\mathbb{Z}}_{p^{2 \times } } {\mathbb{Z}}_{p^{2 \times \cdot \cdot \cdot \times } } {\mathbb{Z}}_{p^2 }

In this paper, we give a construction of partial difference sets in _p ² x _p ² x ... x _p ²using some finite local rings.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThe work of this paper was done when the authors visited the University of Hong Kong.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

Self-dual codes over \mathbb{Z}_8 and \mathbb{Z}_9

We study self-dual codes over the rings and . We define various weights and weight enumerators over these rings and describe the groups of invariants for each weight enumerator over the rings. We examine the torsion codes over these rings to describe the structure of self-dual codes. Finally we classify self-dual codes of small lengths over .     

On some classes of linear codes over $${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$$ and their covering radii

In this paper, we define the simplex and MacDonald codes of types \(\alpha \) and \(\beta \) over \({\mathbb {Z}}_{2}{\mathbb {Z}}_{4}\). We also examine the covering radii of these codes. Further, we study the binary images of these codes and prove that the binary image of the simplex codes of type \(\alpha \) meets the Gilbert bound.     

Modular independence and generator matrices for codes over $${\mathbb {Z}_m}$$

We introduce a new notion of modular independence to define bases and the generator matrices for the codes over the ring of integers of general modulus m. We define standard forms for such generator matrices, and discuss how to find such forms and the parity check matrices.      

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

On real local isometric immersions of {\mathbb{C}Q^2_c} into {\mathbb{C} P^{3}}

We investigate real local isometric immersions of Kähler manifolds ${\mathbb{C}Q^2_c}$ of constant holomorphic curvature 4c into complex projective 3-space. Our main result is that the standard embedding of ${\mathbb{C}P^2}$ into ${\mathbb{C}P^3}$ has strong rigidity under the class of local isometric transformations. We also prove that there are no local isometric immersions of ${\mathbb{C}Q^2_c}$ into ${\mathbb{C}P^3}$ when they have different holomorphic curvature. An important method used is a study of the relationship between the complex structure of any locally isometric immersed ${\mathbb{C}Q^2_c}$ and the complex structure of the ambient space ${\mathbb{C}P^3}$ .     

Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Scheme-Theoretically

We study phylogenetic invariants of general group-based models of evolution with group of symmetries \({\mathbb{Z}_3}\). We prove that complex projective schemes corresponding to the ideal I of phylogenetic invariants of such a model and to its subideal \({I'}\) generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that \({I = I'}\).