The Invariants of the Clifford Groups

The automorphism group of the Barnes-Wall lattice L _m in dimension 2 ^m (m ; 3) is a subgroup of index 2 in a certain Clifford group of structure 2 ₊ ^1+2m . O ⁺(2m,2). This group and its complex analogue of structure .Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for of degree 2k is spanned by the complete weight enumerators of the codes , where C ranges over all binary self-dual codes of length 2k; these are a basis if m k - 1. We also give new constructions for L _m and : let M be the -lattice with Gram matrix . Then L _m is the rational part of M ^m, and = Aut(M^m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then is precisely the automorphism group of the complete weight enumerator of . There are analogues of all these results for the complex group , with doubly-even self-dual code instead of self-dual code.     

Optimal Subcodes of Second Order Reed-Muller Codes andMaximal Linear Spaces of Bivectors of Maximal Rank

There are exactlytwo non-equivalent [32,11,12]-codes in the binaryReed-Muller code which contain and have the weight set {0,12,16,20,32}. Alternatively,the 4-spaces in the projective space over the vector space for which all points have rank 4 fall into exactlytwo orbits under the natural action of PGL(5) on .     

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.     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

Characterizations of Translation Generalized Quadrangles

If x is a regular point of the generalizedquadrangle of order (s,t), s 1 t, then x defines a dual net . If contains a line L of regularpoints and if for at least one point x on Lthe automorphism group of the dual net satisfies certain transitivityproperties, then is a translation generalized quadrangle. Thisresult has many applications. We give one example. Ifs=t 1, then is a dual affine plane. Let be a generalizedquadrangle of orders,s odd and s 1, which contains a lineL of regular points. If for at least one pointx on L the plane is Desarguesian, then is isomorphic to the classical generalizedquadrangleW(s).     

Lexicographic Order and Linearity

Let be a list of all words of , lexicographically ordered with respect to some basis. Lexicodes are codes constructed from by applying a greedy algorithm. A short proof, only based on simple principles from linear algebra, is given for the linearity of these codes. The proof holds for any ordered basis, and for any selection criterion, thus generalizing the results of several authors. An extension of the applied technique shows that lexicodes over are linear for a wide choice of bases and for a large class of selection criteria. This result generalizes a property of Conway and Sloane.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

Equivariant K-Homology and Restriction to Finite Cyclic Subgroups

For a discrete group G, we prove that a G-map between proper G–CW-complexes induces an isomorphism in G-equivariant K-homology if it induces an isomorphism in C-equivariant K-homology for every finite cyclic subgroup C of G. As an application, we show that the source of the Baum–Connes assembly map, namely K _* ^G (E(G, in)), is isomorphic to K _* ^G (E(G, )), where E(G, ) denotes the classifying space for the family of finite cyclic subgroups of G. Letting be the family of virtually cyclic subgroups of G, we also establish that and related results.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

Stickelberger Codes

Let p be an odd prime and be a primitive p th root of unity over . The Galois group G of over is a cyclic group of order p-1. The integral group ring [G] contains the Stickelberger ideal S _p which annihilates the ideal class group of K. In this paper we investigate the parameters of cyclic codes S _p (q) obtained as reductions of S _p modulo primes q which we call Stickelberger codes. In particular, we show that the dimension of S _p (p) is related to the index of irregularity of p, i.e., the number of Bernoulli numbers B _2k , , which are divisible by p. We then develop methods to compute the generator polynomial of S _p (p). This gives rise to anew algorithm for the computation of the index of irregularity of a prime. As an application we show that 20,001,301 is regular. This significantly improves a previous record of 8,388,019 on the largest explicitly known regular prime.     

True Dimension of Some Binary Quadratic Trace Goppa Codes

We compute in this paper the true dimension over of Goppa Codes (L, g) defined by the polynomial proving, this way, a conjecture stated in [14,16].     

Quasivariety Generated by Free Metabelian and 2-Nilpotent Groups

Let qG be a quasivariety generated by a group G and be a non-Abelian quasivariety of groups with a finite lattice of subquasivarieties. Suppose is contained in a quasivariety generated by the following two groups: a free 2-nilpotent group F₂( ₂) of rank 2 and a free metabelian (i.e., with an Abelian commutant) group F₂( ₂) of rank 2. It is proved that either = qF₂( ₂) or = qF₂( ₂) in this instance.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 389–398, July–August, 2005.     

Generalized linear dynamical systems and the classification problem

We consider the classification of generalized linear controllable systems over the field = ℂ or = ℝ under transformations defined by the action of the group GL _n ( ) × GL _n ( ). We review the recent results of Cobb, Helmke, Shayman, Zhou, Hinrichsen, O’Halloran, and others on the geometric structure of the set of orbits C _n,m ( ) of generalized linear controllable systems, which in particular prove smoothness, compactness, and projectivity of C _n,m ( ) and evaluate its dimension. We show that C _n,m ( ) is a natural compactification of the set of orbits of ordinary linear controllable systems ∑ _n,m ( ) and the boundary C _n,m ( ) − ∑ _n,m ( ) consists of the orbits of singular generalized systems. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 153–166, 2004.     

Exponential Stability of Semigroups Related to Operator Models in Mechanics

In this paper, we consider equations of the form , where is a function with values in the Hilbert space , the operator B is symmetric, and the operator A is uniformly positive and self-adjoint in . The linear operator generating the C ₀-semigroup in the energy space is associated with this equation. We prove that this semigroup is exponentially stable if the operator B is uniformly positive and the operator A dominates B in the sense of quadratic forms.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

Let H₁( ) be the usual Hardy space on . We show that the couple (H₁( ), L( ) is a Calderón couple. This result immediately follows from the following stronger one: Given any fH₁( ) +L( ) there exist two linear operators U and V satisfying the properties: (i) Uf=Nf (Nf being the non-tangential maximal function of f) and U is contractive from H₁( ) to L₁( ) and also from L( ) to L( ); (ii) V(Nf)=f, V is similtaneously bounded from L₁( ) to H₁( ) and from L( ) to L( ) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (L_p( ), BMO ( )) for every 1相似文献   

Constructions of Nested Partial Difference Sets with Galois Rings

There have been several recent constructions of partial difference sets (PDSs) using the Galois rings for p a prime and t any positive integer. This paper presents constructions of partial difference sets in where p is any prime, and r and t are any positive integers. For the case where 2$$ " align="middle" border="0"> many of the partial difference sets are constructed in groups with parameters distinct from other known constructions, and the PDSs are nested. Another construction of Paley partial difference sets is given for the case when p is odd. The constructions make use of character theory and of the structure of the Galois ring , and in particular, the ring × . The paper concludes with some open related problems.     

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg then the state complexity of is equal to the Wolf bound. For deg , we use Clifford's theorem to give a simple lower bound on the state complexity of . We then derive two further lower bounds on the state space dimensions of in terms of the gonality sequence of . (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.     

Hyperplane Sections of Kantor's Unitary Ovoids

The projective plane is embedded as a variety of projective points in , where M is a nine dimensional -module for the groupG=GL(3,q ²). The hyperplane sections of thisvariety and their stabilizers in the group G aredetermined. When q 2 (mod 3) one such hyperplanesection is a member of the family of Kantor's unitary ovoids.We furtherdetermine all sections whereD has codimension two in M and demonstratethat these are never empty. Consequences are drawn for Kantor'sovoids.