1,696 research outputs found
Resolvable Mendelsohn designs and finite Frobenius groups
We prove the existence and give constructions of a -fold perfect
resolvable -Mendelsohn design for any integers with such that there exists a finite Frobenius group whose kernel
has order and whose complement contains an element of order ,
where is the least prime factor of . Such a design admits as a group of automorphisms and is perfect when is a
prime. As an application we prove that for any integer in prime factorization, and any prime dividing
for , there exists a resolvable perfect -Mendelsohn design that admits a Frobenius group as a group of
automorphisms. We also prove that, if is even and divides for
, then there are at least resolvable -Mendelsohn designs that admit a Frobenius group as a group of
automorphisms, where is Euler's totient function.Comment: Final versio
Existence of r-fold perfect (v,K,1)-Mendelsohn designs with Kā{4,5,6,7}
AbstractLet v be a positive integer and let K be a set of positive integers. A (v,K,1)-Mendelsohn design, which we denote briefly by (v,K,1)-MD, is a pair (X,B) where X is a v-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t=1,2,ā¦,r, every ordered pair of points of X are t-apart in exactly one block of B, then the (v,K,1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect (v,K,1)-MD. If K={k} and r=kā1, then an r-fold perfect (v,{k},1)-MD is essentially the more familiar (v,k,1)-perfect Mendelsohn design, which is briefly denoted by (v,k,1)-PMD. In this paper, we investigate the existence of r-fold perfect (v,K,1)-Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements
Existence of perfect Mendelsohn designs with k=5 and Ī»>1
AbstractLet Ļ
, k, and Ī» be positive integers. A (Ļ
, k, Ī»)-Mendelsohn design (briefly (Ļ
, k, Ī»)-MD) is a pair (X, B) where X is a Ļ
-set (of points) and B is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair of points of X are consecutive in exactly Ī» blocks of B. A set of k distinct elements {a1, a2,ā¦, ak} is said to be cyclically ordered by a1<a2<āÆ<ak<a1 and the pair ai, ai+t is said to be t-apart in cyclic k-tuple (a1, a2,ā¦, ak) where i+t is taken modulo k. It for all t=1,2,ā¦, k-1, every ordered pair of points of X is t-apart in exactly Ī» blocks of B, then the (Ļ
, k, Ī»)-MD is called a perfect design and is denoted briefly by (Ļ
, k, Ī»)-PMD. In this paper, we shall be concerned mainly with the case where k=5 and Ī»>1. It will be shown that the necessary condition for the existence of a (Ļ
, 5, Ī»)-PMD, namely, Ī»v(Ļ
-1)ā”0 (mod 5), is also sufficient for Ī»>1 with the possible exception of pairs (Ļ
, Ī») where Ī»=5 and Ļ
=18 and 28
Spectrum of Sizes for Perfect Deletion-Correcting Codes
One peculiarity with deletion-correcting codes is that perfect
-deletion-correcting codes of the same length over the same alphabet can
have different numbers of codewords, because the balls of radius with
respect to the Levenshte\u{\i}n distance may be of different sizes. There is
interest, therefore, in determining all possible sizes of a perfect
-deletion-correcting code, given the length and the alphabet size~.
In this paper, we determine completely the spectrum of possible sizes for
perfect -ary 1-deletion-correcting codes of length three for all , and
perfect -ary 2-deletion-correcting codes of length four for almost all ,
leaving only a small finite number of cases in doubt.Comment: 23 page
Uniform hypergraphs containing no grids
A hypergraph is called an rĆr grid if it is isomorphic to a pattern of r horizontal and r vertical lines, i.e.,a family of sets {A1, ..., Ar, B1, ..., Br} such that Aiā©Aj=Biā©Bj=Ļ for 1ā¤i<jā¤r and {pipe}Aiā©Bj{pipe}=1 for 1ā¤i, jā¤r. Three sets C1, C2, C3 form a triangle if they pairwise intersect in three distinct singletons, {pipe}C1ā©C2{pipe}={pipe}C2ā©C3{pipe}={pipe}C3ā©C1{pipe}=1, C1ā©C2ā C1ā©C3. A hypergraph is linear, if {pipe}Eā©F{pipe}ā¤1 holds for every pair of edges Eā F.In this paper we construct large linear r-hypergraphs which contain no grids. Moreover, a similar construction gives large linear r-hypergraphs which contain neither grids nor triangles. For rā„. 4 our constructions are almost optimal. These investigations are motivated by coding theory: we get new bounds for optimal superimposed codes and designs. Ā© 2013 Elsevier Ltd
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