21 research outputs found
From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes
Cages, defined as regular graphs with minimum number of nodes for a given
girth, are well-studied in graph theory. Trapping sets are graphical structures
responsible for error floor of low-density parity-check (LDPC) codes, and are
well investigated in coding theory. In this paper, we make connections between
cages and trapping sets. In particular, starting from a cage (or a modified
cage), we construct a trapping set in multiple steps. Based on the connection
between cages and trapping sets, we then use the available results in graph
theory on cages and derive tight upper bounds on the size of the smallest
trapping sets for variable-regular LDPC codes with a given variable degree and
girth. The derived upper bounds in many cases meet the best known lower bounds
and thus provide the actual size of the smallest trapping sets. Considering
that non-zero codewords are a special case of trapping sets, we also derive
tight upper bounds on the minimum weight of such codewords, i.e., the minimum
distance, of variable-regular LDPC codes as a function of variable degree and
girth
On the existence of combinatorial configurations
A (v, b, r, k) combinatorial configuration can be defined as a connected, (r, k)-biregular bipartite graph with v vertices on one side and b vertices on the other and with no cycle of length 4. Combinatorial configurations have become very important
for some cryptographic applications to sensor networks and to peer-to-peer communities. Configurable tuples are those tuples (v, b, r, k) for which a (v, b, r, k) combinatorial configuration exists.
It is proved in this work that the set of configurable tuples with fixed r and k has the structure of a numerical semigroup.
The semigroup is completely described whenever r = 2 or r = 3.
For the remaining cases some bounds are given on the multiplicity and the conductor of the numerical semigroup. This leads to
some concluding results on the existence of configurable tuples.Peer Reviewe
Bounds for graphs of given girth and generalized polygons
In this paper we present a bound for bipartite
graphs with average bidegrees η and ξ satisfying the inequality η ≥ ξ
α, α ≥ 1. This bound turns out to be the sharpest existing bound.
Sizes of known families of finite generalized polygons are exactly
on that bound. Finally, we present lower bounds for the numbers
of points and lines of biregular graphs (tactical configurations) in
terms of their bidegrees. We prove that finite generalized polygons
have smallest possible order among tactical configuration of given
bidegrees and girth. We also present an upper bound on the size
of graphs of girth g ≥ 2t + 1. This bound has the same magnitude
as that of Erd¨os bound, which estimates the size of graphs without
cycles C₂t