Hyperplane sections of determinantal varieties over finite fields and linear codes

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify the hyperplane sections giving the maximum number of ${\mathbb{F}}_q$-rational points. Further we consider similar questions for sections by linear subvarieties of a fixed codimension in the ambient projective space. This is closely related to the study of linear codes associated to determinantal varieties, and the determination of their weight distribution, minimum distance, and generalized Hamming weights. The previously known results about these are generalized and expanded significantly. Connections to eigenvalues of certain association schemes, distance regular graphs, and rank metric codes are also indicated.     

Split Orthogonal Arrays and Maximum Independent Resilient Systems of Functions

A system of (Boolean) functions in variables is called randomized if the functions preserve the property of their variables to be independent and uniformly distributed random variables. Such a system is referred to as -resilient if for any substitution of constants for any variables, where 0 i t, the derived system of functions in variables will be also randomized. We investigate the problem of finding the maximum number of functions in variables of which any form a -resilient system. This problem is reduced to the minimization of the size of certain combinatorial designs, which we call split orthogonal arrays. We extend some results of design and coding theory, in particular, a duality in bounding the optimal sizes of codes and designs, in order to obtain upper and lower bounds on . In some cases, these bounds turn out to be very tight. In particular, for some infinite subsequences of integers they allow us to prove that , , , , . We also find a connection of the problem considered with the construction of unequal-error-protection codes and superimposed codes for multiple access in the Hamming channel.     

An Infinite Family of 3-Designs from Preparata Codes overZ4

We show that the support of minimum Lee weight codewords having Hamming weight 5 in the Preparata code over Z₄ form a 3-(2^m,5,10) design for any odd integer m 3.     

Two New Infinite Families of 3-Designs from Kerdock Codes over Z4

In this paper we show that the support of the codewords of each type in the Kerdock code of length 2^m over Z₄ form 3-designs for any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer , whose parameters are ,and .     

Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.     

The poset structures admitting the extended binary Golay code to be a perfect code

Brualdi et al. [Codes with a poset metric, Discrete Math. 147 (1995) 57-72] introduced the concept of poset codes, and gave an example of poset structure which admits the extended binary Golay code to be a 4-error-correcting perfect P-code. In this paper we classify all of the poset structures which admit the extended binary Golay code to be a 4-error-correcting perfect P-code, and show that there are no posets which admit the extended binary Golay code to be a 5-error-correcting perfect P-code.     

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, ${{\rm Sp}(2,\mathbb {R})}Let Г be a torsion-free uniform lattice of SU(m, 1), m > 1. Let G be either SU(p, 2) with p ≥ 2, or SO(p, 2) with p ≥ 3. The symmetric spaces associated to these G’s are the classical bounded symmetric domains of rank 2, with the exceptions of SO*(8)/U(4) and SO*(10)/U(5). Using the correspondence between representations of fundamental groups of K?hler manifolds and Higgs bundles we study representations of the lattice Г into G. We prove that the Toledo invariant associated to such a representation satisfies a Milnor-Wood type inequality and that in case of equality necessarily G = SU(p, 2) with p ≥ 2m and the representation is reductive, faithful, discrete, and stabilizes a copy of complex hyperbolic space (of maximal possible induced holomorphic sectional curvature) holomorphically and totally geodesically embedded in the Hermitian symmetric space SU(p, 2)/S(U(p) × U(2)), on which it acts cocompactly.