The Nonexistence of some Griesmer Arcs in PG(4, 5)

In this paper, we prove the nonexistence of arcs with parameters (232, 48) and (233, 48) in PG(4,5). This rules out the existence of linear codes with parameters [232,5,184] and [233,5,185] over the field with five elements and improves two instances in the recent tables by Maruta, Shinohara and Kikui of optimal codes of dimension 5 over F5.

An Improvement to the Achievement of the Griesmer Bound

We denoted by nq(k, d), the smallest value of n for which an [n, k, d]q code exists for given q, k, d. Since nq(k, d) = gq(k, d) for all d ≥ dk + 1 for q ≥ k ≥ 3, it is a natural question whether the Griesmer bound is attained or not for d = dk , where gq(k, d) = ∑[d/q^i], i=0,...,k-1, dk = (k − 2)q^(k−1) − (k − 1)q^(k−2). It was shown by Dodunekov [2] and Maruta [9], [10] that there is no [gq(k, dk ), k, dk ]q code for q ≥ k, k = 3, 4, 5 and for q ≥ 2k − 3, k ≥ 6. The purpose of this paper is to determine nq(k, d) for d = dk as nq(k, d) = gq(k, d) + 1 for q ≥ k with 3 ≤ k ≤ 8 except for (k, q) = (7, 7), (8, 8), (8, 9).

The Nonexistence of [132, 6, 86]3 Codes and [135, 6, 88]3 Codes

We prove the nonexistence of [g3(6, d), 6, d]3 codes for d = 86, 87, 88, where g3(k, d) = ∑⌈d/3i⌉ and i=0 ... k−1. This determines n3(6, d) for d = 86, 87, 88, where nq(k, d) is the minimum length n for which an [n, k, d]q code exists.

Recursive Methods for Construction of Balanced N-ary Block Designs

2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.

Construction of Optimal Linear Codes by Geometric Puncturing

Dedicated to the memory of S.M. Dodunekov (1945–2012)Abstract. Geometric puncturing is a method to construct new codes. ACM Computing Classification System (1998): E.4.

On Projective Plane of Order 13 with a Frobenius Group of Order 39 as a Collineation Group

One of the most outstanding problems in combinatorial mathematics and geometry is the problem of existence of finite projective planes whose order is not a prime power.

Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.

A Relation between the Weyl Group W(e8) and Eight-Line Arrangements on a Real Projective Plane

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

Generalization of a Conjecture in the Geometry of Polynomials

In this paper we survey work on and around the following conjecture, which was first stated about 45 years ago: If all the zeros of an algebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then, for each zero z1 of p, the disk with center z1 and radius r contains at least one zero of the derivative p′ . Until now, this conjecture has been proved for n ≤ 8 only. We also put the conjecture in a more general framework involving higher order derivatives and sets defined by the zeros of the polynomials.

Conference on Geometry and Mathematical Physics, Bulgaria, Zlatograd, 28.08-01.09.2005

List of Participants

When does a Bounded Domain Cover a Projective Manifold? (Survey)

* The research has been partially supported by Bulgarian Funding Organizations, sponsoring the Algebra Section of the Mathematics Institute, Bulgarian Academy of Sciences, a Contract between the Humboldt Univestit¨at and the University of Sofia, and Grant MM 412 / 94 from the Bulgarian Board of Education and Technology

Projective Methods of Image Recognition

We propose a method for image recognition on the base of projections. Radon transform gives an opportunity to map image into space of its projections. Projection properties allow constructing informative features on the base of moments that can be successfully used for invariant recognition. Offered approach gives about 91-97% of correct recognition.

One Application of the Informatics in the Descriptive Geometry

This work acquaints with a program for interactive computer training to students on the subject "Mutual intersecting of pyramids in axonometry ”. Our software is a set of three modules, which we call "student", "teacher" and "autopilot". It gives the final solution of the problem, the traceability of various significant moments in its solution and 3D-image of the finished composition of the two intersecting polyhedra, stripped of the working lines and subjected to rotation and translation.

Two Remarkable Points of the Triangle Geometry

It is shown in the paper the discovery of two remarkable points of the triangle by means of “THE GEOMETER’S SKETCHPAD” software. Some properties of the points are considered too.

Geometry over two Arbitrary Arithmetic Progressions

We consider quadrate matrices with elements of the first row members of an arithmetic progression and of the second row members of other arithmetic progression. We prove the set of these matrices is a group. Then we give a parameterization of this group and investigate about some invariants of the corresponding geometry. We find an invariant of any two points and an invariant of any sixth points. All calculations are made by Maple.