On Sparse Parity Check Matrices

We consider the extremal problem to determine the maximal number of columns of a 0-1 matrix with rows and at most ones in each column such that each columns are linearly independent modulo . For fixed integers and , we shall prove the probabilistic lower bound = ; for a power of , we prove the upper bound which matches the lower bound for infinitely many values of . We give some explicit constructions.     

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).     

Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation of an even codimension is the bundle of the complex structures of the fibers of the transversalbundle of . On there exists a foliation by covering spaces of the leaves of , and any Bottconnection of produces an ordered pair of transversal almost complex structures of . The existence of a Bott connection which yields a structure ₁ that is projectable to the space of leaves isequivalent to the fact that is a transversallyprojective foliation. A Bott connection which yields a projectablestructure ₂ exists iff isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, ₁is projectable as well. ₂ is never integrable.The essential integrability condition of ₁ isthe flatness of the transversal projective structure of .     

On the Nonexistence of q-ary Linear Codes of Dimension Five

There do not exist codes over the Galois field GF attaining the Griesmer bound for for andfor for .     

Primitive Rank 3 Groups on Symmetric Designs

We determine the symmetric designs which admit a group such that G has a nonabelian socle and is a primitiverank 3 group on points (and blocks).     

Frobenius Groups Generated by Quadratic Elements

An automorphism of a group X is said to be quadratic if there exist integers and such that for any . If is a Frobenius group then an element is said to be quadratic if induces, by conjugation in the core of , a quadratic automorphism. By definition, a group H acts on a group F freely if for and only with or . It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in is finite.     

Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

Basic subgroups in abelian group rings

Suppose is a commutative ring with identity of prime characteristic and is an arbitrary abelian -group. In the present paper, a basic subgroup and a lower basic subgroup of the -component and of the factor-group of the unit group in the modular group algebra are established, in the case when is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed -component and of the quotient group are given when is perfect and is arbitrary whose is -divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring is perfect and is -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.     

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.     

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

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.     

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 .     

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.     

Comparison of Sampling Schemes with and without Replacement

An urn contains colored balls, ~balls of each of different colors. The balls are drawn sequentially and equiprobably, one ball at a time, and then each drawn ball drawn is either returned to the urn (sampling with replacement) or left outside the urn (sampling without replacement). The drawing continues until some colors are drawn at least ~times each. Observable statistics are the numbers , , of colors that have appeared precisely ~times each by the stopping time. The asymptotic behavior as of these values for each of the two sampling models is studied; the possibility of their use for identifying the model is discussed.     

Test Rank for Some Free Polynilpotent Groups

We prove a theorem on possible test rank values for groups of the form . It is shown that test rank of a free polynilpotent group is equal to or , for any and every collection of classes. Moreover, for and .     

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

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.     

$${\text{IA}}$$ -Automorphisms of Free Products of Two Abelian Torsion-Free Groups

Let be the free product of two Abelian torsion-free groups, let and , where is the Cartesian subgroup of the group , and let F contain no zero divisors. In the paper it is proved that, in this case, any automorphism of the group is inner. This result generalized the well-known result of Bachmuth, Formanek, and Mochizuki on the automorphisms of groups of the form , , , where is a free group of rank two.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].