1,721,051 research outputs found
Cyclic kite-designs of order v that are cyclically embedded into a cyclic (v,4,2)-design
No full textExplicit constructions for 1-rotational Kirkman triple systems
No full textFor any integer n having each prime factor equivalent to 1 (mod 6) we construct, explicitely, a Kirkman triple system of order 8n+1 admitting an automorphism consisting of a fixed point and a single cycle of length 8n
Partitioned difference families: the storm has not yet passed
Full text linkTwo years ago, we alarmed the scientific community about the large number of bad papers in the literature on zero difference balanced functions, where direct proofs of seemingly new results are presented in an unnecessarily lengthy and convoluted way. Indeed, these results had been proved long before and very easily in terms of difference families. In spite of our report, papers of the same kind continue to proliferate. Regrettably, a further attempt to put the topic in order seems unavoidable. While some authors now follow our recommendation of using the terminology of partitioned difference families, their methods are still the same and their results are often trivial or even wrong. In this note, we show how a very recent paper of this type can be easily dealt with
Super-regular Steiner 2-designs
Full text linkA design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set is contained in G and the elements of each block sum up to zero. The only known Steiner 2-designs that are G-additive for some G have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces AG(n,q), the point-line designs of the projective planes PG(2,q), the point-line designs of the projective spaces PG(n,2) and a sporadic example of a 2-(8191,7,1) design. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly G-additive (the point set is exactly G) and G-regular (any translate of any block is a block as well) at the same time. These designs will be called “G-super-regular”. Our main result is that there are infinitely many values of v for which there exists a super-regular, and therefore additive, 2-(v,k,1) design whenever k is neither singly even nor of the form 2n3≥12. The case k≡2 (mod 4) is a genuine exception whereas k=2n3≥12 is at the moment a possible exception. We also find super-regular 2-(pn,p,1) designs with p∈{5,7} and n≥3 which are not isomorphic to the point-line design of AG(n,p)
Shiftable Heffter spaces
No full textThe shiftable Heffter arrays are naturally generalized to the shiftable Heffter spaces. We present a recursive construction which, starting from a single shiftable Heffter space, leads to infinitely many other shiftable Heffter spaces of the same degree. We also present a direct construction making use of pandiagonal magic squares leading to a shiftable (16 & ell;2,4 & ell;;3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} Heffter space for any & ell;>= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}. Combining these constructions we obtain a shiftable (16 & ell;2mn,4 & ell;n;3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} Heffter space for every triple of positive integers (& ell;,m,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} with m >= n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}
On singular 1-rotational Steiner 2-designs
No full textA Steiner 2-design is 1-rotational over a group G if it admits G as an automorphism group fixing one point and acting regularly on the remainder. 1-rotational Steiner 2-designs have come into fashion since 1981, when Phelps and Rosa (Discrete Math. 33 (1981), 57-66) studied Steiner triple systems that are 1-rotational over the cyclic group. While all 1-rotational Steiner 2-designs constructed in the past have exactly one short block-orbit, in this paper we also consider 1-rotational Steiner 2-designs not having this property. We call them singular and we show that they are quite rare. In particular, we enumerate all
the abelian 1-rotational 2-(49, 4, 1) designs
The 1-rotational Kirkman triple systems of order 33
No full textWe establish that there are exactly 500 KTS(33)s admitting an automorphism group fixing one point and acting regularly on the remainder; 436 are over the cyclic group while 64 are over the dicyclic group. There are exactly 243 nonisomorphic STS(33)s underlying the above KTS(33)s; 211 are over the cyclic group while 32 are over the dicyclic group. This gives a significant improvement on the number of known KTS(33)s (at least 528 instead of at least 28)
Some observations on three classical BIBD constructions
No full textAn examination of three well-known composition techniques allows to get some new information
about the number of non-isomorphic BIBDs with suitable parameters
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