Error-correcting codes are ubiquitous. They are adopted in almost every modern digital communication and storage system, such as wireless communications, optical communications, Flash memories, computer hard drives, sensor networks, and deep-space probing. New-generation and emerging applications demand codes with better error-correcting capability. On the other hand, the design and implementation of those high-gain error-correcting codes pose many challenges. They usually involve complex mathematical computations, and mapping them directly to hardware often leads to very high complexity. VLSI Architectures for Modern Error-Correcting Codes serves as a bridge connecting advancements in coding theory to practical hardware implementations. Instead of focusing on circuit-level design techniques, the book highlights integrated algorithmic and architectural transformations that lead to great improvements on throughput, silicon area requirement, and/or power consumption in the hardware implementation. The goal of this book is to provide a comprehensive and systematic review of available techniques and architectures, so that they can be easily followed by system and hardware designers to develop en/decoder implementations that meet error-correcting performance and cost requirements. This book can be also used as a reference for graduate-level courses on VLSI design and error-correcting coding. Particular emphases are placed on hard- and soft-decision Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes, and binary and non-binary low-density parity-check (LDPC) codes. These codes are among the best candidates for modern and emerging applications due to their good error-correcting performance and lower implementation complexity compared to other codes. To help explain the computations and en/decoder architectures, many examples and case studies are included. More importantly, discussions are provided on the advantages and drawbacks of different implementation approaches and architectures.

Error-correcting codes are ubiquitous. They are adopted in almost every modern digital communication and storage system, such as wireless communications, optical communications, Flash memories, computer hard drives, sensor networks, and deep-space probing. New-generation and emerging applications demand codes with better error-correcting capability. On the other hand, the design and implementation of those high-gain error-correcting codes pose many challenges. They usually involve complex mathematical computations, and mapping them directly to hardware often leads to very high complexity. VLSI Architectures for Modern Error-Correcting Codes serves as a bridge connecting advancements in coding theory to practical hardware implementations. Instead of focusing on circuit-level design techniques, the book highlights integrated algorithmic and architectural transformations that lead to great improvements on throughput, silicon area requirement, and/or power consumption in the hardware implementation. The goal of this book is to provide a comprehensive and systematic review of available techniques and architectures, so that they can be easily followed by system and hardware designers to develop en/decoder implementations that meet error-correcting performance and cost requirements. This book can be also used as a reference for graduate-level courses on VLSI design and error-correcting coding. Particular emphases are placed on hard- and soft-decision Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) codes, and binary and non-binary low-density parity-check (LDPC) codes. These codes are among the best candidates for modern and emerging applications due to their good error-correcting performance and lower implementation complexity compared to other codes. To help explain the computations and en/decoder architectures, many examples and case studies are included. More importantly, discussions are provided on the advantages and drawbacks of different implementation approaches and architectures.

Error correction codes (ECCs) have been widely used in communication systems and storage devices. Nowadays, the rapid development of integrated circuit technologies makes feasible the implementation of powerful ECCs such as turbo code and low-density parity-check (LDPC) code. However, these high-performance codes require complex decoding algorithms, resulting in large hardware area and high power consumption. Furthermore, some of these decoders require an iterative decoding process, which leads to a long decoding latency. Therefore, low-complexity, low-power and high-speed very-large-scale integration (VLSI) architecture design for the ECC decoder is of great importance. This dissertation focuses on efficient VLSI implementation for the decoders of convolutional codes and two advanced coding schemes based on convolutional code: trellis-coded modulation (TCM) and convolutional turbo code (CTC). The first part of this dissertation is dedicated to low-complexity, low-power decoders design for a 4-dimensional, 8-ary phase-shift keying (4-D 8PSK) TCM system. We propose a low-complexity architecture for the transition-metric unit (TMU) to reduce the hardware area without performance loss. Then, a power-efficient scheme by applying T-algorithm on branch metrics (BMs) is proposed for the Viterbi decoder (VD) embedded in the 4-D 8PSK TCM decoder. Unlike the conventional T-algorithm, the proposed scheme does not affect the clock speed of the decoder. Finally, a hybrid T-algorithm is developed by applying T-algorithm on both BMs and path metrics (PMs), which reduces significantly more computations than the conventional T-algorithm applied on PMs. The VLSI design for VDs has been an active research area for decades. In the second part of the dissertation, we extend our research to a more general topic of VDs, where novel architectures are explored to efficiently reduce the power consumption, while still maintaining a high decoding speed and a low decoding latency. CTCs are constructed from parallel convolutional encoding of the same message in different sequences and have the error-correcting capability near the Shannon bound. Practical decoding schemes normally require an iterative decoding process employing the soft-in soft-out (SISO) decoder. The third part of this dissertation is focused on the SISO decoder design for double-binary (DB) CTCs. We propose a low-complexity, memory-reduced architecture by partitioning BMs into two independent portions: information metrics and parity metrics. Furthermore, high-speed recursion architectures for logarithm domain maximum a posteriori probability (log-MAP) algorithm are proposed to increase the decoding speed by algorithmic approximation and bit-level optimization.

This final report provides a brief summary of our research results supported by the above grant during the period from May 1,1998 to November 30, 2001. Our research has addressed design of high-speed, low-energy, low-area architectures for signal processing systems and error control coders. Contributions in the area of error control coding architectures include design of low-energy and low-complexity finite field arithmetic architectures and Reed-Solomon (RS) codecs. High- performance and low-power architectures for low-density parity-check (LDPC) codes have been developed.

Non-binary low-density parity-check (NB-LDPC) codes can achieve better error-correcting performance than binary LDPC codes when the code length is moderate at the cost of higher decoding complexity. The high complexity is mainly caused by the complicated computations in the check node processing and the large memory requirement. In this thesis, two VLSI designs for NB-LDPC decoders based on two novel check node processing schemes are proposed. The first design is based on forward-backward check node processing. A novel scheme and corresponding architecture are developed to implement the elementary step of the check node processing. In our design, layered decoding is applied and only nm less than q messages are kept on each edge of the associated Tanner graph. The computation units and the scheduling of the computations are optimized in the context of layered decoding to reduce the area requirement and increase the speed. This thesis also introduces an overlapped method for the check node processing among different layers to further speed up the decoding. From complexity and latency analysis, our design is much more efficient than any previous design. Our proposed decoder for a (744, 653) code over GF(32) has also been synthesized on a Xilinx Virtex-2 Pro FPGA device. It can achieve a throughput of 9.30 Mbps when 15 decoding iterations are carried out. The second design is based on a proposed trellis based check node processing scheme. The proposed scheme first sorts out a limited number of the most reliable variable-to-check (v-to-c) messages, then the check-to-variable (c-to-v) messages to all connected variable nodes are derived independently from the sorted messages without noticeable performance loss. Compared to the previous iterative forward-backward check node processing, the proposed scheme not only significantly reduced the computation complexity, but eliminated the memory required for storing the intermediate messages generated from the forward and backward processes. Inspired by this novel c-to-v message computation method, we propose to store the most reliable v-to-c messages as 'compressed' c-to-v messages. The c-to-v messages will be recovered from the compressed format when needed. Accordingly, the memory requirement of the overall decoder can be substantially reduced. Compared to the previous Min-max decoder architecture, the proposed design for a (837, 726) code over GF(32) can achieve the same throughput with only 46% of the area.