Orthogonal Designs and Type II Codes over ℤ2k

Symmetric designs are used to construct binary extremal self-dual codes and Hadamard matrices and weighing matrices are used to construct extremal ternary self-dual codes. In this paper, we consider orthogonal designs and related matrices to construct self-dual codes over a larger alphabet. As an example, a number of extremal Type II codes over _2k are constructed.     

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ₆-codes. Then extremal self-dual ₆-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.     

Isodual Codes over ℤ2k and Isodual Lattices

A code is called isodual if it is equivalent to its dual code, and a lattice is called isodual if it is isometric to its dual lattice. In this note, we investigate isodual codes over _2k . These codes give rise to isodual lattices; in particular, we construct a 22-dimensional isodual lattice with minimum norm 3 and kissing number 2464.     

Extremal Type II Z4-codes constructed from binary doubly even self-dual codes of length 40

In this note, we demonstrate that every binary doubly even self-dual code of length $40$ can be realized as the residue code of some extremal Type II ${\mathbb{Z}}_4$-code. As a consequence, it is shown that there are at least $94356$ inequivalent extremal Type II ${\mathbb{Z}}_4$-codes of length $40$.     

A Construction of Cartesian Authentication Codes from Vector Space and Dual Authentication Codes

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Characterizing Extremal Digraphs for Identifying Codes and Extremal Cases of Bondy’s Theorem on Induced Subsets

An identifying code of a (di)graph G is a dominating subset C of the vertices of G such that all distinct vertices of G have distinct (in)neighbourhoods within C. In this paper, we classify all finite digraphs which only admit their whole vertex set as an identifying code. We also classify all such infinite oriented graphs. Furthermore, by relating this concept to a well-known theorem of Bondy on set systems, we classify the extremal cases for this theorem.     

On the Classification of Self-Dual Codes over 𝔽5

 In this paper, we give the classification of self-dual 𝔽₅-codes of lengths 14 and 16. Up to equivalence, there are 53 and 535 such codes, respectively. It is also shown that there is no self-dual [18, 9, 8] code over 𝔽₅. Received: June 18, 2001 Final version received: April 9, 2002 RID="*" ID="*" Supported in part by the Academy of Finland under grants 44517 and 100500     

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

Group of automorphisms of the ring ℤA4

The structure of the group of automorphisms of the integer group ring of the group A₄ is studied in terms of a semidirect product. We show that the Zassenhaus conjecture on the structure of automorphisms of integer group rings of finite groups for the group Aut ?A₄ holds.     

Quasi-localizations of ℤ

Motivated by the categorical notion of localizations applied to the quasi-category of abelian groups, we call a homomorphism α: A → B a quasi-localization of abelian groups if for each ϕ ∈ Hom(A,B) there is an n ∈ ℕ and a unique ψ ∈ End(B) such that nϕ = ψ ∘ α. In this case we call B a quasi-localization of A. In this paper we investigate quasi-localizations of the integers ℤ. While it is well-known that localizations of ℤ are just the E-rings, quasi-localizations of ℤ are much more abundant; an injection α: ℤ → M with M torsion-free, is a quasi-localization if and only if, for R = End(M), one has . We call R the ring of the quasi-localization M. Some old results due to Zassenhaus and Butler show that all rings with free additive groups of finite rank are indeed rings of quasi-localizations of ℤ. We will extend this result and show that there are also rings of infinite rank with this property. While there are many realization results of rings R as endomorphism rings of torsion-free abelian groups M in the literature, the group M is usually not contained in the divisible hull of R ⁺, as is required here. We will use a particular case of a category of left R-modules M with a distinguished family of submodules and thus . We will restrict our discussion to the case M = R such that , and in this case we call the family of left ideals E-forcing, not to be confused with the notion of forcing in set theory. We will provide many examples of quasi-localizations M of ℤ, among them those of infinite rank as well as matrix rings for various rings of finite rank.     

Finitary and Artinian-Finitary Groups over the Integers ℤ

In a series of papers, we have considered finitary (that is, Noetherian-finitary) and Artinian-finitary groups of automorphisms of arbitrary modules over arbitrary rings. The structural conclusions for these two classes of groups are really very similar, especially over commutative rings. The question arises of the extent to which each class is a subclass of the other.Here we resolve this question by concentrating just on the ground ring of the integers . We show that even over neither of these two classes of groups is contained in the other. On the other hand, we show how each group in either class can be built out of groups in the other class. This latter fact helps to explain the structural similarity of the groups in the two classes.     

Abelian groups as UA-modules over the ring ℤ

Let V be a module over a ring R. Themodule V is called a unique addition module (a UA-module) if there is no new addition on the set V without changing the action of R on V. In the paper, the UA-modules over the ring ? are found.     

On the Construction of d-Continuous Modules over Some Special Rings

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Some properties of generalized reduced Verma modules over ℤ-graded modular Lie superalgebras

We study some properties of generalized reduced Verma modules over ?-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and coinduced modules are obtained. Moreover, invariant forms on the generalized reduced Verma modules are considered. In particular, for ?-graded modular Lie superalgebras of Cartan type we prove that generalized reduced Verma modules are isomorphic to mixed products of modules.     

The Existence of Infinitely Many Teichmuller Type Extremal Mappings with Special Boundary Correspondence

in the theory of quasiconformal mappings in the plane, tile problem about whetherthere exist infinitely many Teichmuller type extrmal mappings with special boundarycorrespondence is unsolved so far. In this paper an affirmative answer is given.     

On circular units over the cyclotomic ℤ p -Extension of an abelian field

Let K be an abelian number field of conductor f and p a prime. Sinnott defined circular units C^s_K of K using the norm maps from the cyclotomic units of the nth cyclotomic fields for all n and Washington defined circular units C^w_K of K to be the Galois invariant of the cyclotomic units of the fth cyclotomic field. In this note, we investigate a question raised by Kolster in [5] of whether the projective limits of circular units of Sinnott and Washington over the cyclotomic _p-extension of K are equal. Belliard [2] and Kuera [6] found independently some counter examples to this question. The purpose of this note is to find some conditions on the ground field K under which Kolsters question has an affirmative answer.Mathematics Subject Classification (2000): 11R27, 11R29     

Construction of ‘Wachspress type’ rational basis functions over rectangles

In the present paper, we have constructed rational basis functions ofC ⁰ class over rectangular elements with wider choice of denominator function. This construction yields additional number of interior nodes. Hence, extra nodal points and the flexibility of denominator function suggest better approximation.     

Finite domination and Novikov homology over strongly ℤ-graded rings

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category \(\mathcal{O}\) associated to a reductive finite-dimensional Lie algebra. In particular, we find that, contrary to the original category \(\mathcal{O}\) and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category \(\mathcal{O}\) as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category \(\mathcal{O}\). We subsequently classify all derived equivalence classes of blocks of category \(\mathcal{O}\) in type A which preserve the Koszul grading.     

The distribution of sublattices of ℤ m

Lattices , are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domain of the action ofGL _n () on the generalized upper half-plane _n . Given 1<nm and, letN(D,T) be the number of sublattices of _n which have rankn, similarity class inD, and determinant T. Our most basic result will be thatN(D,T)c ₁(m, n)(D)T ^m asT for suitable setsD, where is the invariant measure on _n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.Herrn Professor Hlawka zum achtzigsten Geburtstag gewidmetSupported in part by NSF-DMS-9401426