32 research outputs found
Mutually orthogonal latin squares with large holes
Two latin squares are orthogonal if, when they are superimposed, every
ordered pair of symbols appears exactly once. This definition extends naturally
to `incomplete' latin squares each having a hole on the same rows, columns, and
symbols. If an incomplete latin square of order has a hole of order ,
then it is an easy observation that . More generally, if a set of
incomplete mutually orthogonal latin squares of order have a common hole of
order , then . In this article, we prove such sets of
incomplete squares exist for all satisfying
Existence of perfect Mendelsohn designs with k=5 and λ>1
AbstractLet υ, k, and λ be positive integers. A (υ, k, λ)-Mendelsohn design (briefly (υ, k, λ)-MD) is a pair (X, B) where X is a υ-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 λ blocks of B. A set of k distinct elements {a1, a2,…, ak} is said to be cyclically ordered by a1<a2<⋯<ak<a1 and the pair ai, ai+t is said to be t-apart in cyclic k-tuple (a1, a2,…, ak) where i+t is taken modulo k. It for all t=1,2,…, k-1, every ordered pair of points of X is t-apart in exactly λ blocks of B, then the (υ, k, λ)-MD is called a perfect design and is denoted briefly by (υ, k, λ)-PMD. In this paper, we shall be concerned mainly with the case where k=5 and λ>1. It will be shown that the necessary condition for the existence of a (υ, 5, λ)-PMD, namely, λv(υ-1)≡0 (mod 5), is also sufficient for λ>1 with the possible exception of pairs (υ, λ) where λ=5 and υ=18 and 28
Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions
This paper examines the construction of low-density parity-check (LDPC) codes
from transversal designs based on sets of mutually orthogonal Latin squares
(MOLS). By transferring the concept of configurations in combinatorial designs
to the level of Latin squares, we thoroughly investigate the occurrence and
avoidance of stopping sets for the arising codes. Stopping sets are known to
determine the decoding performance over the binary erasure channel and should
be avoided for small sizes. Based on large sets of simple-structured MOLS, we
derive powerful constraints for the choice of suitable subsets, leading to
improved stopping set distributions for the corresponding codes. We focus on
LDPC codes with column weight 4, but the results are also applicable for the
construction of codes with higher column weights. Finally, we show that a
subclass of the presented codes has quasi-cyclic structure which allows
low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications
A lower bound on HMOLS with equal sized holes
It is known that , the maximum number of mutually orthogonal latin
squares of order , satisfies the lower bound for large
. For , relatively little is known about the quantity ,
which denotes the maximum number of `HMOLS' or mutually orthogonal latin
squares having a common equipartition into holes of a fixed size . We
generalize a difference matrix method that had been used previously for
explicit constructions of HMOLS. An estimate of R.M. Wilson on higher
cyclotomic numbers guarantees our construction succeeds in suitably large
finite fields. Feeding this into a generalized product construction, we are
able to establish the lower bound for any
and all
The 3-GDDs of type
A 3-GDD of type exists if and only if and have the same parity, divides and .Such a 3-GDD of type is equivalent to an edge decomposition of into triangles
Schröder quasigroups with a specified number of idempotents
AbstractSchröder quasigroups have been studied quite extensively over the years. Most of the attention has been given to idempotent models, which exist for all the feasible orders v, where v≡0,1(mod4) except for v=5,9. There is no Schröder quasigroup of order 5 and the known Schröder quasigroup of order 9 contains 6 non-idempotent elements. It is known that the number of non-idempotent elements in a Schröder quasigroup must be even and at least four. In this paper, we investigate the existence of Schröder quasigroups of order v with a specified number k of idempotent elements, briefly denoted by SQ(v,k). The necessary conditions for the existence of SQ(v,k) are v≡0,1(mod4), 0≤k≤v, k≠v−2, and v−k is even. We show that these conditions are also sufficient for all the feasible values of v and k with few definite exceptions and a handful of possible exceptions. Our investigation relies on the construction of holey Schröder designs (HSDs) of certain types. Specifically, we have established that there exists an HSD of type 4nu1 for u=1,9, and 12 and n≥max{(u+2)/2,4}. In the process, we are able to provide constructions for a very large variety of non-idempotent Schröder quasigroups of order v, all of which correspond to v2×4 orthogonal arrays that have the Klein 4-group as conjugate invariant subgroup
Searching for incomplete self orthogonal latin squares : a targeted and parallel approach
The primary purpose of this dissertation is in the search for new methods
in which to search for Incomplete Self Orthogonal Latin Squares. As such
a full understanding of the structures involved must be examined, starting
from basic Latin Squares. The structures will be explained and built upon in
order to cover Mutually Orthogonal Latin Squares, Frame Latin Squares and
Self Orthogonal Latin Squares. In addition the related structure Orthogonal
Arrays, will be explained as they relate to Incomplete Self Orthogonal Latin
Squares.
This paper also dedicates time to explaining basic search methods and
optimizations that can be done. The two search methods of focus are the
backtracking algorithm and heuristic searches. In our 6nal method the two
will work together to achieve an improved result. The methods currently
being used to search in parallel are also provided, along with the necessary
backup to there structure.
The main research of this paper is focused on the search for Incomplete
Self Orthogonal Squares. This is done by breaking down the problem into
four separate areas of the square. By separating the blocks it enables us to
work on a smaller problem while eliminating many incorrect solutions. The
solution methodology is broken up into three steps and systematically solving
the individual areas of the square.
By taking advantage of the properties of squares to constrain our search as
much as possible we succeeded in reducing the total search time significantly.
Unfortunately, even with our improvement in the overall search time, no open
incomplete self orthogonal latin square problems could be solved. Full results
and comparisons to existing methods are provided