We derive the average symbol and Hamming weight spectrum functions of the random ensemble of regular low-density parity-check (LDPC) codes over GF(2/sup b/) when used with the binary-input noisy channel. This work confirms theoretically that the near-Shannon-limit performance of Gallager's binary LDPC codes can be significantly enhanced by moving to fields of higher order. We construct a family of error-correcting codes based on the binary representation of GF(2/sup b/) codes defined on a cycle Tanner graph that appears to be "good" for both optimum and iterative decoding over the binary-input noisy channel. In particular, we report a short-block-length (1008 bits), rate-1/2 progressive-edge-growth-based cycle Tanner-graph code over GF(2/sup b/) with a block-error rate <10/sup -4/ at E/sub b//N/sub 0/=1.89 dB, which appears to exhibit the best iterative-decoding performance at this short block length known to date.
We derive the average symbol and Hamming weight spectrum functions of the random ensemble of regular low-density parity-check (LDPC) codes over GF(2/sup b/) when used with the binary-input noisy channel. This work confirms theoretically that the near-Shannon-limit performance of Gallager's binary LDPC codes can be significantly enhanced by moving to fields of higher order. We construct a family of error-correcting codes based on the binary representation of GF(2/sup b/) codes defined on a cycle Tanner graph that appears to be "good" for both optimum and iterative decoding over the binary-input noisy channel. In particular, we report a short-block-length (1008 bits), rate-1/2 progressive-edge-growth-based cycle Tanner-graph code over GF(2/sup b/) with a block-error rate <10/sup -4/ at E/sub b//N/sub 0/=1.89 dB, which appears to exhibit the best iterative-decoding performance at this short block length known to date.
The average Hamming weight spectrum function of the random ensemble of regular low-density parity-check (LDPC) codes over GF(2/sup b/) when used with binary-input channels is derived. This work confirms theoretically that the near-Shannon-limit performance of Gallager's binary LDPC codes can be significantly enhanced by moving to fields of higher order. We construct a family of error-correcting codes based on the binary representation of GF(2/sup b/) codes defined on a cycle Tanner graph, that appears to be good for both optimum decoding and iterative decoding over binary-input channels.
This dissertation presents a systematic exposition on finite-block-length coding theory and practice. We begin with the task of identifying the maximum achievable rates over noisy, finite-block-length constrained channels, referred to as (ε, n)-capacity Cεn, with ε denoting target block-error probability and n the block length. We characterize how the optimum codes look like over finite-block-length constrained channels. Constructing good, short-block-length error-correcting codes defined on sparse graphs is the focus of the thesis. We propose a new, general method for constructing Tanner graphs having a large girth by progressively establishing edges or connections between symbol and check nodes in an edge-by-edge manner, called progressive edge-growth (PEG) construction. Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs. The PEG construction attains essentially the same girth as Gallager's explicit construction for regular graphs, both of which meet or exceed an extension of the Erdös-Sachs bound. The PEG construction proves to be powerful for generating good, short-block-length binary LDPC codes. Furthermore, we show that the binary interpretation of GF(2b) codes on the cycle Tanner graph TG(2, dc), if b grows sufficiently large, can be used over the binary-input additive white Gaussian noise (AWGN) channel as "good code for optimum decoding" and "good code for iterative decoding". Codes on sparse graphs are often decoded iteratively by a sum-product algorithm (SPA) with low complexity. We investigate efficient digital implementations of the SPA for decoding binary LDPC codes from both the architectural and algorithmic point of view, and describe new reduced-complexity derivatives thereof. The unified treatment of decoding techniques for LDPC codes provides flexibility in selecting the appropriate design point in high-speed applications from a performance, latency, and computational complexity perspective.
The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number |EJ| of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and |EJ|, one would need to perform the direct enumeration of error patterns with weight i les J. The complexity of enumeration increases exponentially with J, essentially as nJ, where n is the code block length. This limits the application of direct enumeration to codes with small n and J. In this letter, we approximate J and |EJ| by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.
The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number |Ej| of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and |Ej|, one would need to perform the direct enumeration of error patterns with weight i les J. The complexity of enumeration increases exponentially with J, essentially as nJ, where n is the code block length. In this paper, we approximate J and |Ej| by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
Ever since the publication of Shannon's seminal work in 1948, the search for capacity achieving codes has led to many interesting discoveries in channel coding theory. Low-density parity-check (LDPC) codes originally proposed in 1963 were largely forgotten and rediscovered recently. The significance of LDPC codes lies in their capacity approaching performance even when decoded using low complexity sub-optimal decoding algorithms. Iterative decoders are one such class of decoders that work on a graphical representation of a code known as the Tanner graph. Their properties have been well understood in the asymptotic limit of the code length going to infinity. However, the behavior of various decoders for a given finite length code remains largely unknown.An understanding of the failures of the decoders is vital for the error floor analysis of a given code. Broadly speaking, error floor is the abrupt degradation in the frame error rate (FER) performance of a code in the high signal-to-noise ratio domain. Since the error floor phenomenon manifests in the regions not reachable by Monte-Carlo simulations, analytical methods are necessary for characterizing the decoding failures. In this work, we consider hard decision decoders for transmission over the binary symmetric channel (BSC).For column-weight-three codes, we provide tight upper and lower bounds on the guaranteed error correction capability of a code under the Gallager A algorithm by studying combinatorial objects known as trapping sets. For higher column weight codes, we establish bounds on the minimum number of variable nodes that achieve certain expansion as a function of the girth of the underlying Tanner graph, thereby obtaining lower bounds on the guaranteed error correction capability. We explore the relationship between a class of graphs known as cage graphs and trapping sets to establish upper bounds on the error correction capability.We also propose an algorithm to identify the most probable noise configurations, also known as instantons, that lead to error floor for linear programming (LP) decoding over the BSC. With the insight gained from the above analysis techniques, we propose novel code construction techniques that result in codes with superior error floor performance.