4,455 research outputs found
A lower bound for the length of a partial transversal in a latin square
AbstractIt is proved that every n × n Latin square has a partial transversal of length at least n − 5.53(log n)2
Rainbow sets in the intersection of two matroids
Given sets , a {\em partial rainbow function} is a partial
choice function of the sets . A {\em partial rainbow set} is the range of
a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if
and are matroids on the same ground set, and are
pairwise disjoint sets of size belonging to , then there exists a
rainbow set of size belonging to . Following an idea of
Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a
rainbow set of size at least
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 Matroid Generalization of a Result on Row-Latin Rectangles
Let A be an m \times n matrix in which the entries of each row are all
distinct. Drisko showed that, if m \ge 2n-1, then A has a transversal: a set of
n distinct entries with no two in the same row or column. We generalize this to
matrices with entries in a matroid. For such a matrix A, we show that if each
row of A forms an independent set, then we can require the transversal to be
independent as well. We determine the complexity of an algorithm based on the
proof of this result. Lastly, we observe that m \ge 2n-1 appears to force the
existence of not merely one but many transversals. We discuss a number of
conjectures related to this observation (some of which involve matroids and
some of which do not).Comment: 9 pages, 5 figure
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