The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.
The optimal performance-complexity tradeoff for error-correcting codes at rates strictly below the Shannon limit is a central question in coding theory. This paper proposes a numerical approach for the minimization of decoding complexity for long-block-length irregular low-density parity-check (LDPC) codes. The proposed design methodology is applicable to any binary-input memoryless symmetric channel and any iterative message-passing decoding algorithm with a parallel-update schedule. A key feature of the proposed optimization method is a new complexity measure that incorporates both the number of operations required to carry out a single decoding iteration and the number of iterations required for convergence. This paper shows that the proposed complexity measure can be accurately estimated from a density-evolution and extrinsic-information transfer chart analysis of the code. A sufficient condition is presented for convexity of the complexity measure in the variable edge-degree distribution; when it is not satisfied, numerical experiments nevertheless suggest that the local minimum is unique. The results presented herein show that when the decoding complexity is constrained, the complexity-optimized codes significantly outperform threshold-optimized codes at long block lengths, within the ensemble of irregular codes.
The complexity-rate tradeoff for error-correcting codes below the Shannon limit is a central question in coding theory. This paper makes progress in this area by presenting a joint numerical optimization of rate and decoding complexity for low-density parity-check codes. The focus of this paper is on the binary symmetric channel and on a class of decoding algorithms for which an exact extrinsic information transfer (EXIT) chart analysis is possible. This class of decoding algorithms includes the Gallager decoding algorithm B. The main feature of the optimization method is a complexity measure based on the EXIT chart that accurately estimates the number of iterations required for the decoding algorithm to reach a target error rate. Under a fixed check-degree distribution, it is shown that the proposed complexity measure is a convex function of the variable-degree distribution in a region of interest. This allows us to numerically characterize the complexity-rate tradeoff. We show that for the Gallager B decoding algorithm on binary symmetric channels, the optimization procedure can produce complexity savings of 30-40% as compared to the conventional code design method
Time-invariant hybrid (HscrTI) decoding of irregular low-density parity-check (LDPC) codes is studied. Focusing on HscrTIalgorithms with majority-based (MB) binary message-passing constituents, we use density evolution (DE) and finite-length simulation to analyze the performance and the convergence properties of these algorithms over (memoryless) binary symmetric channels. To apply DE, we generalize degree distributions to have the irregularity of both the code and the decoding algorithm embedded in them. A tight upper bound on the threshold of MB HscrTIalgorithms is derived, and it is proven that the asymptotic error probability for these algorithms tends to zero, at least exponentially, with the number of iterations. We devise optimal MB HscrTIalgorithms for irregular LDPC codes, and show that these algorithms outperform Gallager's algorithm A applied to optimized irregular LDPC codes. We also show that compared to switch-type algorithms, such as Gallager's algorithm B, where a comparable improvement is obtained by switching between different MB algorithms, MB HscrTIalgorithms are more robust and can better cope with unknown channel conditions, and thus can be practically more attractive.
By presenting an approximated performance-complexity tradeoff (PCT) algorithm, a low-complexity non-binary low density parity check (LDPC) code over q-ary-input symmetric-output channel is designed in this manuscript which converges faster than the threshold-optimized non-binary LDPC codes in the low error rate regime. We examine our algorithm by both hard and soft decision decoders. Moreover, simulation shows that the approximated PCT algorithm has accelerated the convergence process by 30% regarding the number of the decoding iterations.
Error control coding is an essential part of modern communications systems. LDPC codes have been demonstrated to offer performance near the fundamental limits of channels corrupted by random noise. Optimal maximum likelihood decoding of LDPC codes is too complex to be practically useful even at short block lengths and so a graph-based message passing decoder known as the belief propagation algorithm is used instead. In fact, on graphs without closed paths known as cycles the iterative message passing decoding is known to be optimal and may converge in a single iteration, although identifying the message update schedule which allows single-iteration convergence is not trivial. At finite block lengths graphs without cycles have poor minimum distance properties and perform poorly even under optimal decoding. LDPC codes with large block length have been demonstrated to offer performance close to that predicted for codes of infinite length, as the cycles present in the graph are quite long. In this thesis, LDPC codes of shorter length are considered as they offer advantages in terms of latency and complexity, at the cost of performance degradation from the increased number of short cycles in the graph. For these shorter LDPC codes, the problems considered are: \n \nFirst, improved construction of structured and unstructured LDPC code graphs of short length with a view to reducing the harmful effects of the cycles on error rate performance, based on knowledge of the decoding process. Structured code graphs are particularly interesting as they allow benefits in encoding and decoding complexity and speed. Secondly, the design and construction of LDPC codes for the block fading channel, a particularly challenging scenario from the point of view of error control code design. Both established and novel classes of codes for the channel are considered. Finally the decoding of LDPC codes by the belief propagation algorithm is considered, in particular the scheduling of messages passed in the iterative decoder. A knowledge-aided approach is developed based on message reliabilities and residuals to allow fast convergence and significant improvements in error rate performance.
In this thesis, we propose low-complexity designing algorithms for non-binary low-density
parity-check (NB-LDPC) codes over GF(q) with low error floor and low-complexity decoding
algorithms for binary LDPC codes.
We first propose a low-complexity designing algorithm for for NB-LDPC codes over GF(q)
based on message-passing algorithms. Short cycles in an NB-LDPC code may be even
more harmful to its performance if they do not satisfy the so-called full rank condition
(FRC). This is because they may induce low-weight codewords or absorbing sets in that
case. Thus, it is crucial to count the number of short cycles not satisfying the FRC as well as
the number of short cycles for analyzing the performance of an NB-LDPC code. In order to
count those cycles, we first develop a novel message-passing algorithm and identify how it
is related to the FRC.We then propose a low-complexity algorithmfor counting the number
of short cycles not satisfying the FRC in an NB-LDPC code, as well as the number of short
cycles. The proposed algorithm is not only theoretically validated, but its computational
and storage complexities are also computed. It does not require cycle enumeration with
high complexity, and therefore is still feasible even in the case of large code length, say
≤ 5000, unlike the conventional methods. Based on it, we also provide a locally optimal
rule to choose a nonzero element for a given edge and propose a low-complexity algorithm
for designing an NB-LDPC code with low error floor. Furthermore, several design examples
are presented to verify the validity of the proposed design algorithm.
Secondly, we propose a low-complexity decoding algorithm for binary LDPC codes over
the binary symmetric channel (BSC). The min-sum (MS) algorithm is a low-complexity decoding
algorithm for binary LDPC codes and has lower computational complexity than the
sum-product algorithm (SPA). Since the MS algorithm is an approximation of the SPA, it
has performance degradation. Furthermore, reducing the number of bits for representing
messages to two in a decoding algorithm for further reducing computational and storage
complexities, causes significant performance losses. In order to minimize the performance
degradation and computational and storage complexities, we propose a low-complexity
decoding algorithm which uses several decoders with two-bit precision for binary LDPC
codes over the BSC. The proposed algorithm adaptively changes decoders according to
the number of unsatisfied check nodes at each iteration. Compared with the SPA using
floating-point operations, the proposed decoding algorithm greatly reduces computational
and storage complexities with negligible performance losses. Numerical results show that
the proposed decoding algorithm has similar performance to the SPA with floating-point
operations over the BSC.
In the last part of this thesis, low-complexity decoding algorithms of binary LDPC codes
for the core layer at fixed receivers in ATSC 3.0 are proposed. Gallager’s decoding algorithms
…
We introduce a new one-dimensional (1-D) analysis of low-density parity-check (LDPC) codes on additive white Gaussian noise channels which is significantly more accurate than similar 1-D methods. Our method assumes a Gaussian distribution in message-passing decoding only for messages from variable nodes to check nodes. Compared to existing work, which makes a Gaussian assumption both for messages from check nodes and from variable nodes, our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence. Similar to previous work, the problem of designing irregular LDPC codes reduces to a linear programming problem. However, our method allows irregular code design in a wider range of rates without any limit on the maximum variable-node degree. We use our method to design irregular LDPC codes with rates greater than 1/4 that perform within a few hundredths of a decibel from the Shannon limit. The designed codes perform almost as well as codes designed by density evolution.