Enhanced Error Detection And Correction Codes For Space Communication

Enhanced Error Detection And Correction Codes For Space Communication

¹Dr.Vijayakumar T, ²Hemanth Kumar S, ³Jeevan A T, ⁴Ashwanth M, ⁵Karun Kumar

¹Professor, ²³⁴⁵Student

¹Electronics and Communication,

¹SJB Institute of Technology, Bengaluru, India

¹tvijaykumar@sjbit.edu.in, ²hemanthskumar924@gmail.com, ³jeevanat1210@gmail.com, ⁴imashwanthramayya@gmail.com, ⁵karunkumark800@gmail.com

______________________________________________________________________________________

Abstract— Space communication systems face significant challenges due to harsh channel conditions characterized by high bit error rates, burst errors, and low signal-to-noise ratios. This paper presents an FPGA-based implementation of an enhanced error detection and correction codes for space communication applications. The proposed system integrates CRC-16 error detection with a systematic Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) encoder operating at rate-1/2 with lifting factor Z=16. A block interleaver/deinterleaver pair effectively mitigates burst errors, while an enhanced LDPC decoder employing the offset min-sum algorithm provides robust error correction capabilities. The complete system is successfully implemented on a resource-constrained Xilinx Spartan-6 XC6SLX9 FPGA device. Hardware validation is performed using a 4×4 matrix keypad for data input and a 16×2 LCD display for real-time output visualization. Comprehensive evaluation through simulation waveforms, BER vs SNR analysis, and synthesis reports demonstrates the system's effectiveness in achieving bit error rates below 10⁻³ at 10 dB SNR. Cadence synthesis results show the design occupies 389,391.742 µm² area with 125.2 mW power consumption and a maximum frequency of 66 MHz, validating practical feasibility for satellite communication.

Index Terms— QC-LDPC codes, CRC-16, Block Interleaver, Offset Min-Sum Algorithm, Space Communication, FPGA Implementation, Error Correction, Burst Error Mitigation, Spartan-6, Channel Coding.

_________________________________________________________________________

Introduction

In space communication, reliable data transmission is really a challenging task due to long distances, signal attenuation, cosmic radiation, and intermittent burst errors. Traditional error correction schemes, such as convolutional codes and Reed-Solomon codes, are often inefficient under ultra-low SNR conditions. LDPC codes have emerged as a promising solution, considering their near-Shannon-limit performance. However, their high computational complexity challenges hardware implementation in power- and area-constrained space platforms.

This work presents a hardware-optimized, integrated error detection and correction system that uses CRC-16 for error detection and a QC-LDPC (256,128) code with an Offset Min-Sum decoder for efficient error correction. The system includes a block interleaver to mitigate burst errors and is fully implemented on a Xilinx Spartan-6 XC6SLX9 FPGA. The design is validated through Matlab BER vs SNR simulations, while area, power, and timing metrics are evaluated using the Cadence Genus synthesis tool.

MOTIVATION

The increasing demand for high-data-rate space missions requires dependable communication systems capable of functioning under extreme channel conditions. Existing systems usually make a tradeoff between performance and complexity, resulting in high latency or intolerably high power consumption. This project attempts to fill this gap by providing a low-power, high-performance, FPGA-friendly error correction system suitable for space communications.

OBJECTIVES

• To design and implement a space communication error correction system integrating CRC-16 for error detection, QC-LDPC (256,128) for error correction, and block interleaving for burst error mitigation.
  
• To develop and optimize an LDPC decoder using the Offset Min-Sum algorithm under a realistic channel model with configurable SNR and burst errors.
• To synthesize and deploy the complete system on a Xilinx Spartan-6 XC6SLX9 FPGA, demonstrating real-time functionality with keypad input and LCD output.
• To validate system performance through MATLAB-based BER vs SNR analysis and Cadence synthesis reports for area, power, and timing metrics.

HARDWARE & SOFTWARE REQUIREMENTS

The system is realized on a Xilinx Spartan-6 FPGA (XC6SLX9-CSG324) and is the primary hardware platform for performing real-time error correction. Data input is interfaced using a 4 × 4 keypad, where users can enter 128-bit plaintext in hexadecimal manually. Visual output and real-time visualization of codeword are provided with the aid of a 16 × 2 LCD display, while system monitoring and validation are enabled by a USB-UART interface for serial data logging and debugging.

The software architecture is designed in Verilog HDL for RTL design, using Xilinx ISE 14.7 for the synthesis, place-and-route, and generation of FPGA bit streams. Cadence Genus Synthesis Solution-21.14-s082_1 is used to perform area, power, and timing optimization with a view to utilizing hardware efficiently. Performance validation and channel modeling are made in MATLAB R2024a through BER/FER simulations, error injection based on SNR, and graphical plotting of the performance that characterizes the system against real-world space channel conditions.

BLOCK DIAGRAM OF PROPOSED SYSTEM

Space Channel (SNR & Burst Errors)

CRC-16 Generator

Interleaver Block

QC-LDPC Encoder

Input 128-bit

stores

Noisy 256-bit Codeword

Original

CRC

Compare CRCs crc_match flag

CRC-16 Checker

De-Interleaver Block

Offset Min-Sum Decoder

Decoded 128-bit

1. METHODOLOGY
   
   A. Cyclic Redundancy Check (CRC)

Cyclic Redundancy Check is a widely adopted error detection technique that adds redundancy bits to transmitted data, enabling the receiver to detect errors introduced during transmission. The CRC-16 algorithm generates a 16-bit checksum by treating the data as a polynomial and performing modulo-2 division with a predetermined generator polynomial.

The mathematical representation of CRC computation begins with defining the message polynomial of degree $k-1$:

The generator polynomial for CRC-16-CCITT is:

In binary form, this corresponds to the pattern: $10001000000100001$ (hex: $0x1021$).

To compute the CRC, we first multiply the message polynomial by $x^{16}$ (shifting left by 16 bits):

The CRC remainder $R\left(x\right)$ is obtained via polynomial division in GF(2):

This division is performed using binary arithmetic without carries. The transmitted codeword polynomial is:

At the receiver, the received polynomial $C^{\text{'}}\left(x\right)$ is divided by $G\left(x\right)$:

If $S\left(x\right)=0$, no error is detected (with probability $1-2^{-16}$ for random errors).

Example: For a 128-bit message $M=[10110101\dots \dots \dots \dots .]$, the CRC calculation proceeds as:

B. QC-LDPC Encoder

Low-Density Parity-Check (LDPC) codes are linear block codes characterized by sparse parity-check matrices, offering near-Shannon-limit performance with iterative decoding algorithms. Quasi-Cyclic LDPC (QC-LDPC) codes represent a structured subset where the parity-check matrix $\mathrm{H}$ is composed of circulant permutation matrices or zero matrices, enabling efficient hardware implementation.

For a QC-LDPC code with lifting factor $Z$, the parity-check matrix has the block structure:

$${\mathrm{P}}^b=\{\begin{array}{cc}{\mathrm{I}}^{\left(b\right)}& \mathrm{i}\mathrm{f}b\ge 0\\ 0_{Z\times Z}& \mathrm{i}\mathrm{f}b=-1\end{array}$$where ${\mathrm{I}}^{\left(b\right)}$ is the identity matrix circularly right-shifted by $b$ positions.

Matrix Example: For $Z=4$ and $b=2$:

$${\mathrm{P}}^2=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right]$$

For our implementation with parameters:

• Code rate $R=\frac{K}{N}=\frac{128}{256}=1/2.$
• Lifting factor $Z=16$.
• Information bits $K=128.$
• Codeword length $N=256.$
• Row blocks $m_b=6$.
• Column blocks $n_b=16$.

The base matrix $\mathrm{B}$ is $6\times 16$ with entries from $\left\{-\mathrm{1,0,1},\dots ,15\right\}$:

The systematic codeword is $\mathrm{c}=[\mathrm{s}\mid \mathrm{p}]$, where $\mathrm{s}$ is the information vector (128 bits) and $\mathrm{p}$ is the parity vector (128 bits). The parity-check equation is:

This leads to a system of equations:

where ${\mathrm{c}}_j$ is the $j$-th block of $Z$ bits in the codeword.

The encoding process can be simplified using the block LU decomposition of $\mathrm{H}$. Let $\mathrm{H}=[{\mathrm{H}}_s\mid {\mathrm{H}}_p]$ where ${\mathrm{H}}_s$ corresponds to systematic bits and ${\mathrm{H}}_p$ to parity bits. Then:

C. Block Interleaver

Interleaving disperses burst errors across multiple codewords, converting them into random-like errors that are more easily correctable. For a $Z\times N_B$ block interleaver where $Z$ is the number of rows and $N_B$ is the number of columns.

Let the input sequence be $\mathrm{u}=[u_0,u_1,\dots ,u_{L-1}]$ with $L=Z\times N_B$.

The interleaver writes row-wise into a matrix $\mathrm{U}$ of size $Z\times N_B$:

$${\mathrm{U}}_{i,j}=u_{i\cdot N_B+j},i=0,\dots ,Z-1,j=0,\dots ,N_B-1$$

The interleaved output $\mathrm{v}$ is read column-wise:

Thus, the interleaving mapping function is:

The deinterleaving mapping is:

D. Space Channel Model

The channel model incorporates AWGN and burst errors to simulate space communication conditions. For BPSK modulation, the transmitted signal for bit $x_i\in \left\{\mathrm{0,1}\right\}$ is:

• where $E_s$ is the symbol energy.

The received signal is:

Where,

• $n_i\sim N(0,{\sigma }^2)$ is AWGN with variance ${\sigma }^2=N_0/2$.
• $b_i$ represents burst interference (modeled as a Bernoulli process during bad channel states).

The SNR per bit is:

The bit error probability for BPSK in AWGN is:

For burst errors, we use a Gilbert-Elliott channel model with two states:

• Good state (G): Low error probability $P_g$.
• Bad state (B): High error probability $P_b.$

The state transition probabilities are:

$$P(G\to B)=p,P(B\to G)=q$$

The steady-state probabilities are:

The average error probability is:

For our simulation, typical values are: $P_g={10}^{-5}$, $P_b=0.35$, $p=0.01$, $q=0.1$.

E. QC-LDPC Decoder

The decoder uses the offset min-sum algorithm, a reduced-complexity approximation of the sum-product algorithm.

Let the Tanner graph be defined with:

• $M\left(v\right)$: Set of check nodes connected to variable node $v.$
• $N\left(c\right)$: Set of variable nodes connected to check node $c.$

Initialization:
The channel LLR for variable node $v$ given received value $y_v$ is:

For AWGN channel with BPSK:

Initialize all variable-to-check messages:

Check node Update (CNU):
For each check node $c$ and each neighboring variable node $v$:

where the function $\varphi (\cdot )$ for offset min-sum is:

with offset parameter $\beta$ typically in $[0.5,1.5]$.

Variable Node Update (VNU):

For each variable node $v$ and each neighboring check node $c$:

A posteriori LLR:

Hard decision:

Syndrome check:

If ${\mathrm{s}}^{\left(t\right)}=0$ or $t=T_{\max }$, stop.

1. IMPLEMENTATION

The complete system is implemented and validated through a multi-platform workflow involving simulation, FPGA prototyping, MATLAB analysis, and Cadence Genus Synthesis. 

• 1. Xilinx ISE Implementation & Simulation

The Verilog code is synthesized and simulated using Xilinx ISE 14.7 targeting the Spartan-6 FPGA. The simulation employs a comprehensive testbench that exercises the complete data pipeline through the hierarchical module structure:

The simulation confirms data flow through all pipeline stages—CRC generation, LDPC encoding, interleaving, error injection, deinterleaving, decoding, and CRC verification. The crc_match flag is consistently asserted when the decoded output matches the original input, validating the end-to-end error correction chain under various SNR and burst-error conditions.

• 1. FPGA Prototyping on Spartan-6 Board

The system is deployed on a Xilinx Spartan-6 XC6SLX9 FPGA board with the following top-level hierarchy:

The hardware interfaces includes:

• A 4×4 keypad connected to the keypad_4x4 module for manual hexadecimal input of 128-bit data.
• A 16×2 LCD display driven by the lcd_rg1602a_dynamic controller to show real-time codewords (systematic, parity, corrupted, and decoded bits).
• Tactile switches to control reset, start, burst error enable, SNR selection, and display mode
• LEDs to indicate CRC match/mismatch and system status.

The implementation utilizes 66% of LUTs and 28% of flip-flops on the Spartan-6, with all timing paths meeting the 40 MHz.

• 1. MATLAB Performance Analysis
     
• MATLAB R2024a is used to evaluate the system's coding gain under simulated space-channel conditions.
• A custom script injects AWGN and burst errors into encoded data, runs the LDPC decoder algorithm, and calculates BER and FER across SNR values from 0 dB to 10 Db.
  1. Cadence Genus Synthesis for PPA Metrics

The design is synthesized using Cadence Genus Synthesis Solution (21.14-s082_1) with the command: genus -f run.tcl

The synthesis script targets a slow technology library and extracts area, power, and timing metrics:

• Total Area: 389,391.742 µm² (35,072 cells).
• Total Power: 125.2 mW (92.06% internal, 6.19% switching).
• Timing Slack: –1.454 ns (max frequency = 66 MHz).
  

1. RESULTS AND DISCUSSION
   1. Xilinx Verilog Simulation Waveform

Figure 1: Simulation Waveform

• The simulation waveform confirms the correct functionality of the integrated system—CRC generation, LDPC encoding, interleaving, channel error injection, decoding, and CRC verification.
• As demonstrated in the waveform, the crc_match signal is high (logic '1') when the system operates under moderate SNR conditions, confirming that the decoded output data (data_out) is identical to the original input data (data_in).
• This validation indicates that the LDPC decoder successfully corrected all injected channel errors while the CRC-16 mechanism ensured end-to-end data integrity.
  1. FPGA Implementation Board Output

The hardware implementation on the Spartan-6 FPGA board provides real-time visualization of the error correction system. The physical setup demonstrates practical deployment feasibility.

• enter 32 hexadecimal digits (representing 128 bits) using the 4×4 matrix keypad.
• Each key press generates a 4-bit nibble (0-F).
• Debounce circuitry ensures reliable key detection (10ms debounce period).
• Input accumulates in 128-bit register, displayed progressively on LCD.
• After 32 digits entered, system awaits start trigger.

Figure 2: Input Data Entry via Keypad and Display

• The system is initialized using physical switches: rst_n (reset), enable_burst_errors, snr_db[7:0] (8‑bit SNR setting), and start signal.
• The user enters a 128‑bit hexadecimal value (32 hex digits) via the 4×4 keypad. The entered data (data_in[127:0]) is displayed on the LCD in real‑time, confirming successful input capture and system readiness.
  

Figure 3: Encoded Codeword Display

• The first LCD displays the upper portion of the encoded codeword corresponding to bits [255:128] representing the systematic information bits and upper parity bits generated by the QC-LDPC encoder.
• The second LCD shows the lower portion of the codeword (bits [127:0]) in hexadecimal format, providing complete visibility of the 256-bit encoded result.
• This display mode is activated using the codeword switching key allowing observation of both systematic and parity portions of the rate-1/2 LDPC code.
  

Figure 4: Corrupted Data Display

• The first LCD displays the upper corrupted codeword portion (bits [255:128]) showing visible differences from the original encoded data due to AWGN and burst errors introduced by the space channel model.
• The second LCD presents the lower corrupted portion (bits [127:0]) in hexadecimal format demonstrating how channel impairments degrade the transmitted signal.
• This visualization, enabled through switching keys allows observation of error distribution and validates the channel error injection mechanism at different SNR levels.

Figure 5: Decoded Data Output with CRC Verification

• The LCD displays the 128-bit decoded data (data_out) in hexadecimal format recovered by the enhanced LDPC decoder after processing the corrupted channel output.
• The critical indicator "crc_match = 1" confirms that the CRC-16 checksum computed from decoded output matches the original CRC value, verifying successful error correction.
• The crc_match signal provides definitive end-to-end verification that decoded data is identical to the original input, with crc_match = 1 indicating complete error correction success and crc_match = 0 signaling residual uncorrected errors requiring retransmission.
  1. BER vs SNR Performance
     

Figure 6: Simulation Waveform

Table 1: BER/FER Performance vs. SNR

MATLAB simulations were conducted to evaluate the system’s Bit Error Rate (BER) performance under varying Signal-to-Noise Ratio (SNR) conditions, and the resulting table represents the corresponding values of Uncoded BER, Coded BER, Coding Gain (dB), and Frame Error Rate (FER) for each SNR level.

• 1. RTL Schematic Diagram
     
     
     
     Figure 7: RTL Schematic from ISE

The Register Transfer Level (RTL) schematic generated by Xilinx ISE illustrates the structural hierarchy and interconnections of the synthesized design. The hierarchical view reveals:

• Top-Level Structure (enhanced_space_ldpc_system): The schematic shows the main processing module with clearly defined input/output ports.
• Input Ports : clk, rst_n, data_in, start, snr_db[7:0], enable_burst_errors.
• Output Ports: data_out[127:0], codeword_out[255:0], corrupted_data_out[255:0], decoded_codeword_out[255:0], valid_out, bit_errors[7:0], crc_original[15:0], crc_decoded[15:0], crc_match, burst_errors[7:0], total_errors[7:0] and decoder_iterations[5:0].

The RTL view confirms the modular hierarchical design approach, with well-defined interfaces enabling independent module development and testing.

• 1. Design Summary and FPGA Resource Utilization

FPGA Resource Utilization (Xilinx Spartan-6 XC6SLX9):

Table 2: Design Summary Table

Key Resource Observations:

• LUT utilization (66%) is the primary constraint.
• Block RAM usage modest (12.5%) - efficient LLR storage.
• No DSP slices used - all arithmetic in fabric logic.
• Sufficient margin for future enhancements.
• I/O usage high due to LED/LCD interfaces.
  
  1. Cadence Area, Power, and Timing Reports
     

1. Area Report

The synthesis using Cadence Genus Synthesis Solution 21.14-s082_1 provides detailed area breakdown:

Table 3: Cadence Area Report

Detailed Analysis: The total design occupies 389,391.742 µm² in the target technology library comprising:

Enhanced Decoder Module (82.01%):

• Dominates the design area with 319,305.832 µm².
• Contains LLR memory arrays (L_ch, L_app, R_msg).
• Check node processing units with min-finding logic.
• Variable node accumulation arithmetic.
• Control FSM and iteration management.

Remaining Modules (18.00%):

• QC-LDPC Encoder: =12% (ROM for base matrix + shift logic).
• Space Channel: =3% (LFSR generators + corruption logic).
• CRC Modules (2 instances): =2% (LFSR + polynomial logic).
• Interleaver/Deinterleaver: =1% (combinational wiring).
  

1. Power Report

Power analysis performed at typical operating conditions:

Table 4: Cadence Power Report

Total Power Consumption (125.2 mW):

• Leakage Power: 2.194 mW (1.75%).
• Internal Power: 115.3 mW (92.06%).
• Switching Power: 7.750 mW (6.19%).

Register Power (86.97% of total = 108.9 mW):

• Dominates power consumption.
• LLR storage registers in decoder.
• State registers for FSMs.
• Data path registers for pipelining.
• High activity due to iterative decoding.

Logic Power (13.03% of total = 16.31 mW):

• Combinational logic gates.
• Arithmetic units (adders, comparators, min-finding).
• Control logic and multiplexers.
• Relatively low due to limited gate switching.

Power Distribution Analysis:

• Internal power (92.06%): Clock-related power in sequential elements.
• Switching power (6.19%): Data-dependent signal transitions.
• Leakage power (1.75%): Static power in modern technology nodes.|
  

1. Timing Report

Table 5: Cadence Timing Report

1. FUTURE SCOPE
2. Scalability to Larger Codewords: Extend the design to support higher code rates (e.g., 3/4) and longer block lengths (e.g., 1024 bits) to improve throughput and error correction performance for deep-space missions.
3. Advanced FPGA Platform Migration: Migrate to modern FPGA families like Xilinx Artix-7 or Kintex-7, or even space-grade radiation-tolerant FPGAs, to leverage higher logic density, embedded DSP blocks, and improved power efficiency.
4. Multi-Channel and MIMO Support: Extend the architecture to support multiple input multiple output (MIMO) systems for increased data rates and robustness in satellite networks.
5. AI/ML-Enhanced Decoding: Explore machine learning-aided LDPC decoding techniques to reduce iterations and power consumption while maintaining or improving BER performance.
6. Conclusion

“This system integrates CRC-16 for error detection, QC-LDPC encoding/decoding, block interleaving, and the Offset Min-Sum algorithm to provide reliable error correction under low-SNR and burst-error space-channel conditions. It is fully implemented and validated on the Xilinx Spartan-6 XC6SLX9 FPGA, with performance verified through Matlab BER vs SNR simulations. This system’s hardware efficiency is further confirmed using Cadence Genus synthesis, which provides detailed reports on area, power and timing”.

1. References
2. R. G. Gallager, "Low-Density Parity-Check Codes," IRE Transactions on Information Theory, vol. 8, no. 1, pp. 21-28, Jan. 1962.
3. D. J. C. MacKay and R. M. Neal, "Near Shannon Limit Performance of Low Density Parity Check Codes," Electronics Letters, vol. 33, no. 6, pp. 457-458, Mar. 1997.
4. T. Richardson and R. Urbanke, "The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding," IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 599-618, Feb. 2001.
5. M. P. C. Fossorier, M. Mihaljevic, and H. Imai, "Reduced Complexity Iterative Decoding of Low-Density Parity Check Codes Based on Belief Propagation," IEEE Transactions on Communications, vol. 47, no. 5, pp. 673-680, May 1999.
6. J. Chen, A. Dholakia, E. Eleftheriou, M. P. C. Fossorier, and X.-Y. Hu, "Reduced-Complexity Decoding of LDPC Codes," IEEE Transactions on Communications, vol. 53, no. 8, pp. 1288-1299, Aug. 2005.
7. Z. Li, L. Chen, L. Zeng, S. Lin, and W. H. Fong, "Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes," IEEE Transactions on Communications, vol. 54, no. 1, pp. 71-81, Jan. 2006.
8. CCSDS, "TM Synchronization and Channel Coding," Recommendation for Space Data System Standards, CCSDS 131.0-B-3, Blue Book, Sept. 2017.
9. ETSI EN 302 307-1, "Digital Video Broadcasting (DVB); Second Generation Framing Structure, Channel Coding and Modulation Systems for Broadcasting, Interactive Services, News Gathering and Other Broadband Satellite Applications; Part 1: DVB-S2," v1.4.1, Nov. 2014.
10. A. I. Vila Casado, M. Griot, and R. D. Wesel, "Informed Dynamic Scheduling for Belief-Propagation Decoding of LDPC Codes," IEEE Transactions on Communications, vol. 55, no. 12, pp. 2278-2287, Dec. 2007.
11. K. Zhang, X. Huang, and Z. Wang, "High-Throughput Layered Decoder Implementation for Quasi-Cyclic LDPC Codes," IEEE Journal on Selected Areas in Communications, vol. 27, no. 6, pp. 985-994, Aug. 2009.
12. J. Zhao, F. Zarkeshvari, and A. H. Banihashemi, "On Implementation of Min-Sum Algorithm and Its Modifications for Decoding Low-Density Parity-Check (LDPC) Codes," IEEE Transactions on Communications, vol. 53, no. 4, pp. 549-554, Apr. 2005.
13. S. Lin and D. J. Costello, Error Control Coding, 2nd ed., Upper Saddle River, NJ, USA: Prentice-Hall, 2004.
14. T. K. Moon, Error Correction Coding: Mathematical Methods and Algorithms, Hoboken, NJ, USA: Wiley-Interscience, 2005.
15. B. Sklar, Digital Communications: Fundamentals and Applications, 2nd ed., Upper Saddle River, NJ, USA: Prentice-Hall, 2001.
16. F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, "Factor Graphs and the Sum-Product Algorithm," IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498-519, Feb. 2001.
17. H. Zhong, W. Xu, N. Xie, and T. Zhang, "Area-Efficient Min-Sum Decoder Design for High-Rate Quasi-Cyclic Low-Density Parity-Check Codes in Magnetic Recording," IEEE Transactions on Magnetics, vol. 43, no. 12, pp. 4117-4122, Dec. 2007.
18. Y. Chen and K. K. Parhi, "Overlapped Message Passing for Quasi-Cyclic Low-Density Parity Check Codes," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 51, no. 6, pp. 1106-1113, June 2004.
19. W. E. Ryan and S. Lin, Channel Codes: Classical and Modern, Cambridge University Press, 2009.
20. Xilinx Inc., "Spartan-6 Family Overview," DS160 (v2.0), Oct. 2011.
21. P. G. Gulak, "VLSI Structures for LDPC Decoders," in Proc. IEEE Workshop on Signal Processing Systems (SiPS), pp. 1-6, Oct. 2004.
22. E. N. Gilbert, "Capacity of a Burst-Noise Channel," Bell System Technical Journal, vol. 39, no. 5, pp. 1253-1265, Sept. 1960.
23. E. O. Elliott, "Estimates of Error Rates for Codes on Burst-Noise Channels," Bell System Technical Journal, vol. 42, no. 5, pp. 1977-1997, Sept. 1963.
24. M. Eroz, F.-W. Sun, and L.-N. Lee, "DVB-S2 Low Density Parity Check Codes With Near Shannon Limit Performance," International Journal of Satellite Communications and Networking, vol. 22, no. 3, pp. 269-279, May 2004.
25. G. C. Clark Jr. and J. B. Cain, Error-Correction Coding for Digital Communications, New York: Plenum Press, 1981.
26. J. L. Ramsey, "Realization of Optimum Interleavers," IEEE Transactions on Information Theory, vol. 16, no. 3, pp. 338-345, May 1970.