17 research outputs found
An Efficient Algorithm for Finding Dominant Trapping Sets of LDPC Codes
This paper presents an efficient algorithm for finding the dominant trapping
sets of a low-density parity-check (LDPC) code. The algorithm can be used to
estimate the error floor of LDPC codes or to be part of the apparatus to design
LDPC codes with low error floors. For regular codes, the algorithm is initiated
with a set of short cycles as the input. For irregular codes, in addition to
short cycles, variable nodes with low degree and cycles with low approximate
cycle extrinsic message degree (ACE) are also used as the initial inputs. The
initial inputs are then expanded recursively to dominant trapping sets of
increasing size. At the core of the algorithm lies the analysis of the
graphical structure of dominant trapping sets and the relationship of such
structures to short cycles, low-degree variable nodes and cycles with low ACE.
The algorithm is universal in the sense that it can be used for an arbitrary
graph and that it can be tailored to find other graphical objects, such as
absorbing sets and Zyablov-Pinsker (ZP) trapping sets, known to dominate the
performance of LDPC codes in the error floor region over different channels and
for different iterative decoding algorithms. Simulation results on several LDPC
codes demonstrate the accuracy and efficiency of the proposed algorithm. In
particular, the algorithm is significantly faster than the existing search
algorithms for dominant trapping sets
Refined Upper Bounds on Stopping Redundancy of Binary Linear Codes
The -th stopping redundancy of the binary
code , , is defined as the minimum number of rows in
the parity-check matrix of , such that the smallest stopping set is
of size at least . The stopping redundancy is defined as
. In this work, we improve on the probabilistic analysis of
stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best
bounds known today. In our approach, we judiciously select the first few rows
in the parity-check matrix, and then continue with the probabilistic method. By
using similar techniques, we improve also on the best known bounds on
, for . Our approach is compared to the
existing methods by numerical computations.Comment: 5 pages; ITW 201
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
Trapping Sets of Quantum LDPC Codes
Iterative decoders for finite length quantum low-density parity-check (QLDPC)
codes are attractive because their hardware complexity scales only linearly
with the number of physical qubits. However, they are impacted by short cycles,
detrimental graphical configurations known as trapping sets (TSs) present in a
code graph as well as symmetric degeneracy of errors. These factors
significantly degrade the decoder decoding probability performance and cause
so-called error floor. In this paper, we establish a systematic methodology by
which one can identify and classify quantum trapping sets (QTSs) according to
their topological structure and decoder used. The conventional definition of a
TS from classical error correction is generalized to address the syndrome
decoding scenario for QLDPC codes. We show that the knowledge of QTSs can be
used to design better QLDPC codes and decoders. Frame error rate improvements
of two orders of magnitude in the error floor regime are demonstrated for some
practical finite-length QLDPC codes without requiring any post-processing.Comment: Revised version - 19 pages, 12 figures - Accepted for publication in
Quantu