138 research outputs found
A note on strong Skolem starters
In 1991, Shalaby conjectured that any additive group , where
or 3 (mod 8) and , admits a strong Skolem starter and
constructed these starters of all admissible orders . Only
finitely many strong Skolem starters have been known. Recently, in [O.
Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong Skolem Starters},
J. Combin. Des. {\bf 27} (2018), no. 1, 5--21] was given an infinite families
of them. In this note, an infinite family of strong Skolem starters for
, where mod 8 is a prime integer, is presented
A note on two orthogonal totally -free one-factorizations of complete graphs
A pair of orthogonal one-factorizations and of
the complete graph is totally -free, if the union , for any
, does not include a cycle of length four.
In this note, we prove if (mod 4) is a prime power with ,
then there is a pair of orthogonal totally -free one-factorizations of
New constructions of strongly regular Cayley graphs on abelian groups
Davis and Jedwab (1997) established a great construction theory unifying many
previously known constructions of difference sets, relative difference sets and
divisible difference sets. They introduced the concept of building blocks,
which played an important role in the theory. On the other hand, Polhill (2010)
gave a construction of Paley type partial difference sets (conference graphs)
based on a special system of building blocks, called a covering extended
building set, and proved that there exists a Paley type partial difference set
in an abelian group of order for any odd positive integer and
any . His result covers all orders of nonelementary abelian groups in
which Paley type partial difference sets exist. In this paper, we give new
constructions of strongly regular Cayley graphs on abelian groups by extending
the theory of building blocks. The constructions are large generalizations of
Polhill's construction. In particular, we show that for a positive integer
and elementary abelian groups , , of order such
that , there exists a decomposition of the complete graph on the
abelian group by strongly regular
Cayley graphs with negative Latin square type parameters , where and . Such strongly
regular decompositions were previously known only when or is a
-group. Moreover, we find one more new infinite family of decompositions of
the complete graphs by Latin square type strongly regular Cayley graphs. Thus,
we obtain many strongly regular graphs with new parameters.Comment: 14 pages; Some typos are fixed, and Abst and Intro are rewritte
Quaternary Constant-Composition Codes with Weight Four and Distances Five or Six
The sizes of optimal constant-composition codes of weight three have been
determined by Chee, Ge and Ling with four cases in doubt. Group divisible codes
played an important role in their constructions. In this paper, we study the
problem of constructing optimal quaternary constant-composition codes with
Hamming weight four and minimum distances five or six through group divisible
codes and Room square approaches. The problem is solved leaving only five
lengths undetermined. Previously, the results on the sizes of such quaternary
constant-composition codes were scarce.Comment: 23 pages, 3 table
Strong Skolem Starters
This paper concerns a class of combinatorial objects called Skolem starters,
and more specifically, strong Skolem starters, which are generated by Skolem
sequences.
In 1991, Shalaby conjectured that any additive group , where
or , admits a strong Skolem starter and
constructed these starters of all admissible orders . Only
finitely many strong Skolem starters have been known to date.
In this paper, we offer a geometrical interpretation of strong Skolem
starters and explicitly construct infinite families of them.Comment: 17 pages, 2 figure
On strong Skolem starters
In this note we present an alternative (simple) construction of cardioidal
starters (strong and Skolem) for , where (mod 8)
is an odd prime number and is an integer number; also for
and , for infinitely many odd primes
(mod 8) and an integer number. This cardioidal starters
can be obtained by a more general result given by Ogandzhanyants et al. [O.
Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong Skolem starters},
J. Combin. Des. {\bf 27} (2018), no. 1, 5--21]
On strong Skolem starters for
In 1991, N. Shalaby conjectured that any additive group , where
or 3 (mod 8) and , admits a strong Skolem starter and
constructed these starters of all admissible orders . Shalaby
and et al. [O. Ogandzhanyants, M. Kondratieva and N. Shalaby, \emph{Strong
Skolem Starters}, J. Combin. Des. {\bf 27} (2018), no. 1, 5--21] was proved if
, where is a prime number such that
(mod 4) and is a non-negative integer, for
all , then admits a strong Skolem starter. On the
other hand, the author [A. V\'azquez-\'Avila, \emph{A note on strong Skolem
starters}, Discrete Math. Accepted] gives different families of strong Skolem
starters for than Shalaby et al, where (mod 8) is an
odd prime. Recently, the author [A. V\'azquez-\'Avila, \emph{New families of
strong Skolem starters}, Submitted] gives different families of strong Skolem
starters of than Shalaby et al, where (mod 8) and
is an integer greater than 1.
In this paper, we gives some different families of strong Skolem starters of
, where (mod 8) are prime numbers such that
and .Comment: arXiv admin note: substantial text overlap with arXiv:1907.0526
Old and new families of strong Skolem Starters
In this paper, we give new families of strong Skolem starters for
and , for infinitely many odd primes
(mod 8) and be an integer.Comment: The paper gives an alternative (simple) construction of strong Skolem
starters for and , for infinitely many
odd primes (mod 8) and be an integer, than given by
Ogandzhanyants et al. [O. Ogandzhanyants, M. Kondratieva, and N. Shalaby,
\emph{Strong Skolem starters}, J. Combin. Des. {\bf 27} (2018), no. 1, 5--21
Biproducts and Kashina's examples
We revisit a class of examples described in the original paper on biproducts,
expand the class, and provide a detailed analysis of the coalgebra and algebra
structures of many of these examples. Connections with the semisimple Hopf
algebras of dimension a power of two determined by Kashina are examined.
The finite-dimensional non-trivial semisimple cosemisimple Hopf algebras we
construct are shown to be lower cosolvable. Some of these have one proper
normal Hopf subalgebra and are not lower solvable
Semi-cyclic holey group divisible designs with block size three and applications to sampling designs and optical orthogonal codes
We consider the existence problem for a semi-cyclic holey group divisible
design of type (n,m^t) with block size 3, which is denoted by a 3-SCHGDD of
type (n,m^t). When t is odd and n\neq 8 or t is doubly even and t\neq 8, the
existence problem is completely solved; when t is singly even, many infinite
families are obtained. Applications of our results to two-dimensional balanced
sampling plans and optimal two-dimensional optical orthogonal codes are also
discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1304.328
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