Minimum-Integer Computation Finite Alphabet Message Passing Decoder: From Theory to Decoder Implementations towards 1 Tb/s

Abstract

In Message Passing (MP) decoding of Low-Density Parity Check (LDPC) codes, extrinsic information is exchanged between Check Node (CNs) and Variable Node (VNs). In a practical implementation, this information exchange is limited by quantization using only a small number of bits. In recent investigations, a novel class of Finite Alphabet Message Passing (FA-MP) decoders are designed to maximize the Mutual Information (MI) using only a small number of bits per message (e.g., 3 or 4 bits) with a communication performance close to high-precision Belief Propagation (BP) decoding. In contrast to the conventional BP decoder, operations are given as discrete-input discrete-output mappings which can be described by multidimensional LUT (mLUTs). A common approach to avoid exponential increases in the size of mLUTs with the node degree is given by the sequential LUT (sLUT) design approach, i.e., by using a sequence of two-dimensional Lookup-Table (LUTs) for the design, leading to a slight performance degradation. Recently, approaches such as Reconstruction-Computation-Quantization (RCQ) and Mutual Information-Maximizing Quantized Belief Propagation (MIM-QBP) have been proposed to avoid the complexity drawback of using mLUTs by using pre-designed functions that require calculations over a computational domain. It has been shown that these calculations are able to represent the mLUT mapping exactly by executing computations with infinite precision over real numbers. Based on the framework of MIM-QBP and RCQ, the Minimum-Integer Computation (MIC) decoder design generates low-bit integer computations that are derived from the Log-Likelihood Ratio (LLR) separation property of the information maximizing quantizer to replace the mLUT mappings either exactly or approximately. We derive a novel criterion for the bit resolution that is required to represent the mLUT mappings exactly. Furthermore, we show that our MIC decoder has exactly the communication performance of the corresponding mLUT decoder, but with much lower implementation complexity. We also perform an objective comparison between the state-of-the-art Min-Sum (MS) and the FA-MP decoder implementations for throughput towards 1 Tb/s in a state-of-the-art 28 nm Fully-Depleted Silicon-on-Insulator (FD-SOI) technology. Furthermore, we demonstrate that our new MIC decoder implementation outperforms previous FA-MP decoders and MS decoders in terms of reduced routing complexity, area efficiency and energy efficiency.

High-Throughput Polar Code Decoders with Information Bottleneck Quantization

“…From an implementation perspective, the LUTs are transformed into Boolean functions. Despite the logic optimization executed by state-of-the-art synthesis tools, the resulting logic costs can quickly outweigh the benefits of reduced bit widths, particularly for increasing LUT sizes [ 8 ].…”

“…This yields a non-uniform quantization. While IB-based quantization for Low Density Parity Check (LDPC) decoder implementations is well investigated [ 6 , 7 , 8 ], the efficiency of the IB method for polar code decoders is quite unexplored. It is an open research question whether IB-based quantization in polar decoders can yield more efficient implementations compared to standard quantization methods.…”

“…It was shown in [ 8 , 9 ] that a pure lookup table (LUT)-based application of the IB method yields a very large LUTs, making this approach unfeasible. The Reconstruction Computation Quantization (RCQ) scheme [ 10 ] also uses a LUTs to reconstruct and, after computation, quantize back to a smaller bit width.…”