19,050 research outputs found
Intercalates and Discrepancy in Random Latin Squares
An intercalate in a Latin square is a Latin subsquare. Let be
the number of intercalates in a uniformly random Latin square. We
prove that asymptotically almost surely
, and that
(therefore
asymptotically almost surely for any ). This
significantly improves the previous best lower and upper bounds. We also give
an upper tail bound for the number of intercalates in two fixed rows of a
random Latin square. In addition, we discuss a problem of Linial and Luria on
low-discrepancy Latin squares
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
Low-Density Arrays of Circulant Matrices: Rank and Row-Redundancy Analysis, and Quasi-Cyclic LDPC Codes
This paper is concerned with general analysis on the rank and row-redundancy
of an array of circulants whose null space defines a QC-LDPC code. Based on the
Fourier transform and the properties of conjugacy classes and Hadamard products
of matrices, we derive tight upper bounds on rank and row-redundancy for
general array of circulants, which make it possible to consider row-redundancy
in constructions of QC-LDPC codes to achieve better performance. We further
investigate the rank of two types of construction of QC-LDPC codes:
constructions based on Vandermonde Matrices and Latin Squares and give
combinatorial expression of the exact rank in some specific cases, which
demonstrates the tightness of the bound we derive. Moreover, several types of
new construction of QC-LDPC codes with large row-redundancy are presented and
analyzed.Comment: arXiv admin note: text overlap with arXiv:1004.118
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