Channel coding lies at the heart of digital communication and data storage, and this detailed introduction describes the core theory as well as decoding algorithms, implementation details, and performance analyses. In this book, Professors Ryan and Lin provide clear information on modern channel codes, including turbo and low-density parity-check (LDPC) codes. They also present detailed coverage of BCH codes, Reed-Solomon codes, convolutional codes, finite geometry codes, and product codes, providing a one-stop resource for both classical and modern coding techniques. Assuming no prior knowledge in the field of channel coding, the opening chapters begin with basic theory to introduce newcomers to the subject. Later chapters then extend to advanced topics such as code ensemble performance analyses and algebraic code design. 250 varied and stimulating end-of-chapter problems are also included to test and enhance learning, making this an essential resource for students and practitioners alike.

Motivated by the need to communicate short control messages in 5G and beyond, this paper carefully designs codes for cyclic redundancy check (CRC)-aided list decoding of tail-biting convolutional codes (TBCCs) and polar codes. Both codes send a 32-bit message using an 11-bit CRC and 512 transmitted bits. We aim to provide a careful, fair comparison of the error performance and decoding complexity of polar and TBCC techniques for a specific case. Specifically, a TBCC is designed to match the rate of a (512, 43) polar code, and optimal 11-bit CRCs for both codes are designed. The paper examines the distance spectra of the polar and TBCC codes, illuminating the different distance structures for the two code types. We consider both adaptive and non-adaptive CRC-aided list decoding schemes. For polar codes, an adaptive decoder must start with a larger list size to avoid an error floor. For rate-32/512 codes with an 11-bit CRC, the optimized CRC-TBCC design achieves a lower total failure rate than the optimized CRC-polar design. Simulations showed that the optimized CRC-TBCC design achieved significantly higher throughput than the optimized CRC-polar design, so that the TBCC solution achieved a lower total failure rate while requiring less computational complexity.

We extend earlier work on the design of convolutional code-specific CRC codes to Q-ary alphabets, with an eye toward Q-ary orthogonal signaling. Starting with distance-spectrum optimal, zero-terminated, Q-ary convolutional codes, we design Q-ary CRC codes so that the CRC/convolutional concatenation is distance-spectrum optimal. The Q-ary code symbols are mapped to a Q-ary orthogonal signal set and sent over an AWGN channel with noncoherent reception. We focus on Q = 4, rate-1/2 convolutional codes in our designs. The random coding union bound and normal approximation are used in earlier works as benchmarks for performance for distance-spectrum-optimal convolutional codes. We derive a saddlepoint approximation of the random coding union bound for the coded noncoherent signaling channel, as well as a normal approximation for this channel, and compare the performance of our codes to these limits. Our best design is within 0.6 dB of the RCU bound at a frame error rate of 10(-4).

This paper explores list decoding of convolutional and polar codes for short messages such as those found in the 5G physical broadcast channel. A cyclic redundancy check (CRC) is used to select a codeword from a list of likely codewords. One example in the 5G standard encodes a 32-bit message with a 24-bit CRC and a 512-bit polar code with additional bits added by repetition to achieve a very low rate of 32/864. This paper shows that optimizing the CRC length improves the E b /N 0 performance of this polar code, where E b /N 0 is the ratio of the energy per data bit to the noise power spectral density. Furthermore, even better E b / N 0 performance is achieved by replacing the polar code with a tail-biting convolutional code (TBCC) with a distance-spectrum-optimal (DSO) CRC. This paper identifies the optimal CRC length to minimize the frame error rate (FER) of a rate-1/5 TBCC at a specific value of E b / N 0 . We also show that this optimized TBCC/CRC can attain the same excellent E b / N 0 performance with the very low rate of 32/864 of the 5G polar code, where the low rate is achieved through repetition. We show that the proposed TBCC/CRC concatenated code outperforms the PBCH polar code described in the 5G standard both in terms of FER and decoding run time. We also explore the tradeoff between undetected error rate and erasure rate as the CRC size varies.

This paper presents designs and constructions of two classes of binary quasi-cyclic LDPC codes for correcting multiple random phased-bursts of erasures over the binary erasure channel. The erasure correction of codes in both classes is characterized by the cycle and adjacency structure of their Tanner graphs. Erasure correction of these codes is a very simple process which requires only modulo-2 additions. The codes in the second class are capable of correcting locally and globally distributed phased-bursts of erasures with a two-phase iterative erasure-correction process.

This paper presents some new results on QC-LDPC codes constructed based on Reed-Solomon (RS) codes. Results include designs and constructions of RS-based QC-LDPC codes with girth 8, cycle structure in their Tanner graphs, and correction of erasures.

In order to adapt to the ever-increasing demands of telecommunication needs, todays network operators are implementing 100 Gb/s per dense wavelength division multiplexing (DWDM) channel transmission. At those data rates, the performance of fiberoptic communication systems is degraded significantly due to intra- and inter-channel fiber nonlinearities, polarization-mode dispersion (PMD), and chromatic dispersion. In order to deal with those channel impairments, novel advanced techniques in modulation and detection, coding and signal processing are needed. This unique book represents a coherent and comprehensive introduction to the fundamentals of optical communications, signal processing and coding for optical channels. It is the first to integrate the fundamentals of coding theory with the fundamentals of optical communication.

Protograph-based LDPC and generalized LDPC (G-LDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. The design of such “protograph-based codes” relies on what is effectively a computer-based search. As such, following Gallager, it is prudent to restrict the search to a “good ensemble,” for example, an ensemble whose minimum distance grows linearly with codeword length. A good ensemble can also mean one with good stopping set, trapping set, or pseudocodeword properties. In this paper, ensemble codeword weight enumerators for finite-length LDPC and G-LDPC codes based on protographs were derived, and then the asymptotic case was considered. The asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length. Then, the codeword weight enumerator technique is adapted to yield ensemble stopping set, trapping set, and pseudocodeword enumerators for protograph LDPC and G-LDPC codes. In this case, the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size, trapping set size, and pseudoweight grows linearly with codeword length. Trapping set enumerators for G-LDPC code ensembles represent a more complex problem which we do not consider here.

Protograph-based generalized LDPC (GLDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. Recently, a technique for computing ensemble weight enumerators for GLDPC codes has been published. In the current paper, we investigate the existence of typical minimum distance for protograph-based GLDPC codes. That is, we first upper bound the ensemble weight enumerators for finite-length GLDPC codes based on protographs, and then we consider the sum of weight enumerators. The results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. We provide conditions on the connections of degree-2 variable nodes to constraint nodes (short block codes) to have typical minimum distance. These conditions are related to the minimum distances of the constraint nodes.

Preface 1. Coding and capacity 2. Finite fields, vector spaces, finite geometries and graphs 3. Linear block codes 4. Convolutional codes 5. Low-density parity-check codes 6. Computer-based design of LDPC codes 7. Turbo codes 8. Ensemble enumerators for turbo and LDPC codes 9. Ensemble decoding thresholds for LDPC and turbo codes 10. Finite geometry LDPC codes 11. Constructions of LDPC codes 12. LDPC codes based on combinatorial designs, graphs, and superposition 13. LDPC codes for binary erasure channels 14. Non-binary LDPC codes 15. LDPC code applications and advanced topics Index.

Two-dimensional magnetic recording (TDMR) is a novel storage architecture that, in theory, can achieve a density of up to 10 Tb/in2. It uniquely differs from other proposed next-generation architectures because of its reliance on sophisticated 2-D signal-processing algorithms. Recently, a number of contributions have been made in the development of read-channel models and detectors for TDMR systems. In this paper, we provide a detailed review on all important read-channel models under consideration. Our discussion focuses on the purpose of each model, placing a special emphasis on the suitability of the Voronoi model for the purpose of designing detectors. We also propose several detection schemes for TDMR based on the Voronoi model and present some numerical results.

Recently, pseudocodewords of Tanner graphs of LDPC codes have been used to explain the behavior of iterative decoders for these codes. In this paper, finite-length pseudocodeword weight enumerators for protograph-based generalized-LDPC code ensembles are obtained. Then asymptotic results are derived from the finite-length results by letting the block length go to infinity. The asymptotic results allow us to determine whether or not the typical minimum pseudocodeword weight grows linearly with codeword length. We give examples with Hamming component codes.

One of the most significant impediments to the use of LDPC codes in many communication and storage systems is the error-rate floor phenomenon associated with their iterative decoders. The error floor has been attributed to certain subgraphs of an LDPC codepsilas Tanner graph induced by so-called trapping sets. We show in this paper that once we identify the trapping sets of an LDPC code of interest, a sum-product algorithm (SPA) decoder can be custom-designed to yield floors that are orders of magnitude lower than the conventional SPA decoder. We present three classes of such decoders: (1) a bi-mode decoder, (2) a bit-pinning decoder which utilizes one or more outer algebraic codes, and (3) three generalized-LDPC decoders. We demonstrate the effectiveness of these decoders for two codes, the rate-1/2 (2640,1320) Margulis code which is notorious for its floors and a rate-0.3 (640,192) quasi-cyclic code which has been devised for this study. Although the paper focuses on these two codes, the decoder design techniques presented are fully generalizable to any LDPC code.