Related papers: Group divisible designs with block size five: dire…
For the category of group codes, that generalizes the category of linear codes over a finite field, and with the generalized notions of direct sums and ndecomposable group codes, we prove that every MDS non trivial code, every perfect non…
A strong $s$-blocking set in a projective space is a set of points that intersects each codimension-$s$ subspace in a spanning set of the subspace. We present an explicit construction of such sets in a $(k - 1)$-dimensional projective space…
The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal…
We present some constructions of groupoids as: direct product, semidirect product, and we give necessary and sufficient conditions for a groupoid to be embedded into a direct product of groupoids. Also, we establish necessary and sufficient…
Two product array codes are used to construct the (24, 12, 8) binary Golay code through the direct sum operation. This construction provides a systematic way to find proper (8, 4, 4) linear block component codes for generating the Golay…
The purpose of this paper is to give explicit constructions of unitary $t$-designs in the unitary group $U(d)$ for all $t$ and $d$. It seems that the explicit constructions were so far known only for very special cases. Here explicit…
A group code structure of a linear code is a description of the code as one-sided or two-sided ideal of a group algebra of a finite group. In these realizations, the group algebra is identified with the ambient space, and the group elements…
This paper presents a novel construction method for symmetric Sudoku-type games based on Lee distance perfect codes and diameter perfect codes. The proposed method utilizes the tiling property of these codes to define the structure of the…
Many generator matrices for constructing extremal binary self-dual codes of different lengths have the form G=(I|A), where I is the n by n identity matrix and A is the n by n matrix fully determined by the first row. In this work, we define…
Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified.
In this paper, we present an explicit construction of list-decodable codes for single-deletion and single-substitution with list size two and redundancy 3log n+4, where n is the block length of the code. Our construction has lower…
We define a triangle design as a partition of the set of lines of a projective space into triangles, where a triangle consists of three pairwise intersecting lines with no common point. A triangle design is balanced if all points are…
This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine spaces for large $t$, with a focus on the important index $\lambda=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive…
Given positive integers $v$, $k$, $t$ and $\lambda$ with $v \geq k \geq t$, a packing design PD$_{\lambda}(v,k,t)$ is a pair $(V,\mathcal{B})$, where $V$ is a $v$-set and $\mathcal{B}$ is a collection of $k$-subsets of $V$ such that each…
Let G be a regular Lie group which is a directed union of regular Lie groups G_i (all modelled on possibly infinite-dimensional, locally convex spaces). We show that G is the direct limit of the G_i as a regular Lie group whenever G admits…
Block-transitive Steiner $t$-designs form a central part of the study of highly symmetric combinatorial configurations at the interface of several disciplines, including group theory, geometry, combinatorics, coding and information theory,…
Problem 21 of Brauer's list of problems from 1963 asks whether for any positive integer k there are finitely many isomorphism classes of groups that occur as the defect group of a block with k irreducible characters. We solve this problem…
We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs,…
The aim of this paper is to see how commuting graphs interact with two semigroup constructions: the zero-union and the direct product. For both semigroup constructions, we investigate the diameter, clique number, girth, chromatic number and…
Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely…