This document discusses cyclic codes, which are a type of error correcting code used in digital communications. Cyclic codes have the property that any cyclic shift of a codeword is also a valid codeword. They are defined by a generator polynomial that is a factor of x^n + 1, where n is the codeword length. Cyclic codes allow for simple encoding and decoding circuits using shift registers. Examples of cyclic codes include repetition codes, Hamming codes, BCH codes, and Reed-Solomon codes.
This document discusses error control coding, also known as channel coding. It begins by introducing concepts such as bit error rate and methods to reduce it, including increasing transmission power, diversity techniques, automatic repeat request (ARQ), and forward error correction (FEC) codes. It then provides details on parity checks, cyclic redundancy checks (CRCs), and block error correction codes. Specific error control coding techniques like Reed-Solomon codes are also mentioned.
The International Journal of Engineering and Science (The IJES)theijes
This document summarizes a research paper that presents a unified hybrid Reed-Solomon decoder architecture capable of correcting both burst errors and random errors/erasures. The architecture combines low-complexity algorithms for correcting burst errors and random errors. It first provides background on Reed-Solomon codes, including their encoding and standard decoding process. It then describes the proposed unified hybrid decoding architecture, which uses a reformulated inversionless algorithm for burst error correction and integrates it with standard algorithms like Berlekamp-Massey for random error correction. The architecture is the first to allow multi-mode Reed-Solomon decoding to handle different error types.
The document describes Reed-Solomon error correcting codes. It begins by defining error correcting codes and their use of redundancy to recover corrupted data. It then discusses encoding messages into codewords by evaluating polynomials over finite fields and transmitting the codewords. The relationship between code rate, relative distance and error correction capability is also covered, along with the Singleton bound. Reed-Solomon codes specifically encode messages as polynomial coefficients and evaluate the polynomials at predefined points to generate codewords.
Convolution codes - Coding/Decoding Tree codes and Trellis codes for multiple...Madhumita Tamhane
In contrast to block codes, Convolution coding scheme has an information frame together with previous m information frames encoded into a single code word frame, hence coupling successive code word frames. Convolution codes are most important Tree codes that satisfy certain additional linearity and time invariance properties. Decoding procedure is mainly devoted to correcting errors in first frame. The effect of these information symbols on subsequent code word frames can be computed and subtracted from subsequent code word frames. Hence in spite of infinitely long code words, computations can be arranged so that the effect of earlier frames, properly decoded, on the current frame is zero.
This document discusses channel coding and linear block codes. Channel coding adds redundant bits to input data to allow error detection and correction at the receiver. Linear block codes divide the data into blocks, encode each block into a larger codeword, and use a generator matrix to map message blocks to unique codewords. The codewords can be detected and sometimes corrected using a parity check matrix. Hamming codes are a type of linear block code that can correct single bit errors. The document provides examples of encoding data using generator matrices and decoding using syndrome values and parity check matrices. It also discusses how the minimum distance of a code determines its error detection and correction capabilities.
This document discusses Reed-Solomon error correcting codes. It begins with an introduction to Reed-Solomon codes and their use in communication and data storage. It then provides details on Reed-Solomon encoding and decoding. The decoding process involves calculating syndromes, finding error locations using the Chien search algorithm, and determining error values using Forney's algorithm. Extensions of the inversionless Massey-Berlekamp algorithm are also described, which can compute the error locator and evaluator polynomials simultaneously without field inversions.
Linear block codes take binary data in blocks and encode them into longer codewords by adding redundant bits to allow for error detection and correction. The document discusses key concepts of linear block codes including generator and parity check matrices, syndrome detection, minimum distance, and applications. It also provides an example of a (6,3) linear block code and a (7,4) Hamming code.
This document discusses error control coding, also known as channel coding. It begins by introducing concepts such as bit error rate and methods to reduce it, including increasing transmission power, diversity techniques, automatic repeat request (ARQ), and forward error correction (FEC) codes. It then provides details on parity checks, cyclic redundancy checks (CRCs), and block error correction codes. Specific error control coding techniques like Reed-Solomon codes are also mentioned.
The International Journal of Engineering and Science (The IJES)theijes
This document summarizes a research paper that presents a unified hybrid Reed-Solomon decoder architecture capable of correcting both burst errors and random errors/erasures. The architecture combines low-complexity algorithms for correcting burst errors and random errors. It first provides background on Reed-Solomon codes, including their encoding and standard decoding process. It then describes the proposed unified hybrid decoding architecture, which uses a reformulated inversionless algorithm for burst error correction and integrates it with standard algorithms like Berlekamp-Massey for random error correction. The architecture is the first to allow multi-mode Reed-Solomon decoding to handle different error types.
The document describes Reed-Solomon error correcting codes. It begins by defining error correcting codes and their use of redundancy to recover corrupted data. It then discusses encoding messages into codewords by evaluating polynomials over finite fields and transmitting the codewords. The relationship between code rate, relative distance and error correction capability is also covered, along with the Singleton bound. Reed-Solomon codes specifically encode messages as polynomial coefficients and evaluate the polynomials at predefined points to generate codewords.
Convolution codes - Coding/Decoding Tree codes and Trellis codes for multiple...Madhumita Tamhane
In contrast to block codes, Convolution coding scheme has an information frame together with previous m information frames encoded into a single code word frame, hence coupling successive code word frames. Convolution codes are most important Tree codes that satisfy certain additional linearity and time invariance properties. Decoding procedure is mainly devoted to correcting errors in first frame. The effect of these information symbols on subsequent code word frames can be computed and subtracted from subsequent code word frames. Hence in spite of infinitely long code words, computations can be arranged so that the effect of earlier frames, properly decoded, on the current frame is zero.
This document discusses channel coding and linear block codes. Channel coding adds redundant bits to input data to allow error detection and correction at the receiver. Linear block codes divide the data into blocks, encode each block into a larger codeword, and use a generator matrix to map message blocks to unique codewords. The codewords can be detected and sometimes corrected using a parity check matrix. Hamming codes are a type of linear block code that can correct single bit errors. The document provides examples of encoding data using generator matrices and decoding using syndrome values and parity check matrices. It also discusses how the minimum distance of a code determines its error detection and correction capabilities.
This document discusses Reed-Solomon error correcting codes. It begins with an introduction to Reed-Solomon codes and their use in communication and data storage. It then provides details on Reed-Solomon encoding and decoding. The decoding process involves calculating syndromes, finding error locations using the Chien search algorithm, and determining error values using Forney's algorithm. Extensions of the inversionless Massey-Berlekamp algorithm are also described, which can compute the error locator and evaluator polynomials simultaneously without field inversions.
Linear block codes take binary data in blocks and encode them into longer codewords by adding redundant bits to allow for error detection and correction. The document discusses key concepts of linear block codes including generator and parity check matrices, syndrome detection, minimum distance, and applications. It also provides an example of a (6,3) linear block code and a (7,4) Hamming code.
A second important technique in error-control coding is that of convolutional coding . In this type of coding the encoder output is not in block form, but is in the form of an encoded
sequence generated from an input information sequence.
convolutional encoding is designed so that its decoding can be performed in some structured and simplified way. One of the design assumptions that simplifies decoding
is linearity of the code. For this reason, linear convolutional codes are preferred. The source alphabet is taken from a finite field or Galois field GF(q).
Convolution coding is a popular error-correcting coding method used in digital communications.
The convolution operation encodes some redundant information into the transmitted signal, thereby improving the data capacity of the channel.
Convolution Encoding with Viterbi decoding is a powerful FEC technique that is particularly suited to a channel in which the transmitted signal is corrupted mainly by AWGN.
It is simple and has good performance with low implementation cost.
The document discusses coding and error control coding. It defines coding as a procedure that maps messages into encoded messages to improve communication efficiency. Error control coding adds redundant bits to messages to allow detection and correction of errors during transmission. Channel encoders add redundant bits systematically while channel decoders can detect and correct errors in the received information bits using the redundant bits. Common error control methods are forward error correction and error detection with retransmission. Block codes and convolution codes are examples of error control codes discussed. Key concepts like codewords, minimum distance, and conditions for error detection and correction are also summarized.
1) The document discusses various concepts related to image compression including why it is needed, different types of images and sources, lossy vs lossless compression, and entropy coding methods like Huffman coding and predictive coding.
2) Key concepts covered include the need for compression due to large file sizes of images, different image types and sources, the tradeoff between lossy and lossless compression, and how entropy coding assigns codes based on probability distributions to reduce redundancy.
3) Different coding techniques are described like Huffman coding which creates a variable length code table based on probabilities in a bottom-up approach, and predictive coding which encodes prediction errors rather than raw pixel values to remove spatial redundancy.
This document summarizes a lecture on entropy coding and discusses Hoffman coding and Golomb coding. It begins with an overview of entropy, conditional entropy, and mutual information. It then explains Hoffman coding by describing the Hoffman coding procedure and properties like optimality. Golomb coding is also summarized, including the Golomb code construction and its advantages over unary coding. Implementation details are provided for Golomb encoding and decoding.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
Lecture 4 from virtual university of pakistanSaba Hanif
The document discusses error detecting and correcting techniques used in wireless networks. It reviews parity checks, cyclic redundancy checks (CRC), and block error correction codes. CRC uses a predetermined polynomial and modulo-2 arithmetic to generate a checksum over transmitted data blocks. This allows the receiver to detect errors by comparing the received checksum. Block error correction codes add redundancy to transmitted data blocks to allow the receiver to detect and correct a certain number of bit errors based on the code's Hamming distance.
High Speed Decoding of Non-Binary Irregular LDPC Codes Using GPUs (Paper)Enrique Monzo Solves
Implementation of a high speed decoding of non-binary irregular LDPC codes using CUDA GPUs.
Moritz Beermann, Enrique Monzó, Laurent Schmalen, Peter Vary
IEEE SiPS, Oct. 2013, Taipei, Taiwan
This document provides an overview of coding theory and recent advances in low-density parity-check (LDPC) codes. It discusses Shannon's channel coding theorem and how modern error-correcting codes achieve rates close to channel capacity. LDPC codes are described as having sparse parity-check matrices and being decoded iteratively using message passing. The performance of LDPC codes can be analyzed using density evolution and threshold calculations. Linear programming decoding is introduced as an alternative decoding approach that has connections to message passing decoding.
This document provides information about BCH codes, including:
1. BCH codes are linear cyclic block codes that can detect and correct errors. They allow flexibility in choosing block length and code rate.
2. Key characteristics of BCH codes include the block length being 2m - 1, error correction ability up to t errors where t<(2m - 1)/2, and minimum distance of at least 2t + 1.
3. Galois fields are finite fields that are important for constructing BCH codes. A generator polynomial is chosen based on the roots in the Galois field and is used to encode messages into codewords.
The document proposes a new normalization technique for compensating the optimistic output information produced by a soft output Viterbi algorithm (SOVA) decoder in a turbo decoder. The technique counts the sign differences between the a-priori and extrinsic information to determine a normalization factor for each data block. Simulations show the proposed technique achieves about a 0.2 dB coding gain improvement on average compared to other techniques, while reducing the number of iterations needed for decoding by up to 21.
This document discusses error detection and correction techniques at the data link layer. It covers different types of errors, the use of redundancy to detect or correct errors, block coding and convolutional coding approaches. Specific coding schemes like parity checks, cyclic redundancy checks (CRC), and Hamming codes are explained in detail. The key aspects covered are the use of redundant bits, minimum Hamming distance requirements for detection and correction capabilities, and how techniques like CRC and Hamming codes function to detect and correct single-bit errors. Assignments and example problems are also listed.
The document outlines Reed-Solomon error correction codes. It discusses how Reed-Solomon codes encode data using a generator polynomial to produce parity check symbols. The document then describes how Reed-Solomon codes can decode errors using syndrome calculation, error location polynomials, and finding the error positions and values through algorithms like Forney's method and Chien search. Reed-Solomon codes are widely used in applications like CDs, DVDs, wireless communications and digital television for their ability to efficiently correct both random and burst errors.
This document discusses error detection and correction techniques at the data link layer. It covers various types of errors that can occur and how redundancy is used to detect or correct them. Error correcting codes like block codes and convolutional codes are introduced. Specific coding schemes like parity checks, cyclic redundancy checks (CRC), and Hamming codes are explained in detail. The document provides examples of how these codes are implemented and their performance characteristics in terms of detecting and correcting single and burst errors. Standard polynomials used in CRC and properties of good polynomials are also discussed.
Convolutional codes are a type of error correcting code where each coded output block depends not only on the corresponding input block but also on previous input blocks. The encoder contains shift registers and logic gates. Convolutional codes are characterized by parameters k (input bits), n (output bits), and m (memory order). The distance properties and performance of convolutional codes depends on the constraint length L, which is a function of k and m. Convolutional codes can be represented using trees, trellises, state diagrams or polynomials to describe the encoding process.
This document discusses convolutional codes. It begins by defining a convolutional code as an error-correcting code that transforms each m-bit information symbol into an n-bit symbol, where the code rate is m/n. It then provides an example of a (2,1,8) convolutional coder with specific generator sequences. It includes diagrams of the coder circuit and trellis and discusses encoding an example message sequence. It also provides questions and answers demonstrating encoding and decoding operations using MATLAB.
The document describes experiments to generate and demodulate various digital modulation schemes using MATLAB, including:
- ASK modulation and demodulation using an envelope detector.
- BPSK modulation by changing the phase of a carrier signal and demodulation using correlation.
- FSK modulation by changing the frequency of a carrier signal and demodulation using correlation and subtraction.
- QPSK modulation using four phases to transmit two bits per symbol and gray encoding of the dibits.
The document describes experiments on generating and demodulating amplitude shift keying (ASK), phase shift keying (PSK), and frequency shift keying (FSK) signals using MATLAB.
For ASK, binary data is used to modulate a carrier signal by varying its amplitude. Demodulation recovers the data using an envelope detector. For PSK, binary data modulates the phase of a carrier signal. Demodulation correlates the signal with a reference carrier. For FSK, binary data determines the frequency of the carrier signal. Demodulation correlates the signal with two reference carriers and compares the results.
The MATLAB program for each modulation scheme generates test signals, plots the results, and recovers the data
Proof of Kraft Mc-Millan theorem - nguyen vu hungVu Hung Nguyen
This document provides a proof of the Kraft-McMillan theorem, which establishes a necessary condition for uniquely decodable codes. It begins by defining the Kraft inequality and proving that if a code C is uniquely decodable, the sum of 2 to the power of the negative codeword lengths must be less than or equal to 1. It then shows that if this condition is not met, the nth power of K(C) cannot grow linearly, proving the inequality holds. The document concludes by constructing a prefix code that satisfies a given set of codeword lengths meeting the Kraft inequality condition.
This document presents the design and implementation of an FPGA-based BCH decoder. It discusses BCH codes, which are binary error-correcting codes used in wireless communications. The implemented decoder is for a (15, 5, 3) BCH code, meaning it can correct up to 3 errors in a block of 15 bits. The decoder uses a serial input/output architecture and is implemented using VHDL on a FPGA device. It performs BCH decoding through syndrome calculation, running the Berlekamp-Massey algorithm to solve the key equation, and using Chien search to find error locations. The simulation result verifies correct decoding operation.
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
A second important technique in error-control coding is that of convolutional coding . In this type of coding the encoder output is not in block form, but is in the form of an encoded
sequence generated from an input information sequence.
convolutional encoding is designed so that its decoding can be performed in some structured and simplified way. One of the design assumptions that simplifies decoding
is linearity of the code. For this reason, linear convolutional codes are preferred. The source alphabet is taken from a finite field or Galois field GF(q).
Convolution coding is a popular error-correcting coding method used in digital communications.
The convolution operation encodes some redundant information into the transmitted signal, thereby improving the data capacity of the channel.
Convolution Encoding with Viterbi decoding is a powerful FEC technique that is particularly suited to a channel in which the transmitted signal is corrupted mainly by AWGN.
It is simple and has good performance with low implementation cost.
The document discusses coding and error control coding. It defines coding as a procedure that maps messages into encoded messages to improve communication efficiency. Error control coding adds redundant bits to messages to allow detection and correction of errors during transmission. Channel encoders add redundant bits systematically while channel decoders can detect and correct errors in the received information bits using the redundant bits. Common error control methods are forward error correction and error detection with retransmission. Block codes and convolution codes are examples of error control codes discussed. Key concepts like codewords, minimum distance, and conditions for error detection and correction are also summarized.
1) The document discusses various concepts related to image compression including why it is needed, different types of images and sources, lossy vs lossless compression, and entropy coding methods like Huffman coding and predictive coding.
2) Key concepts covered include the need for compression due to large file sizes of images, different image types and sources, the tradeoff between lossy and lossless compression, and how entropy coding assigns codes based on probability distributions to reduce redundancy.
3) Different coding techniques are described like Huffman coding which creates a variable length code table based on probabilities in a bottom-up approach, and predictive coding which encodes prediction errors rather than raw pixel values to remove spatial redundancy.
This document summarizes a lecture on entropy coding and discusses Hoffman coding and Golomb coding. It begins with an overview of entropy, conditional entropy, and mutual information. It then explains Hoffman coding by describing the Hoffman coding procedure and properties like optimality. Golomb coding is also summarized, including the Golomb code construction and its advantages over unary coding. Implementation details are provided for Golomb encoding and decoding.
This document contains solved problems related to digital communication systems. It begins by defining key elements of digital communication systems such as source coding, channel encoders/decoders, and digital modulators/demodulators. It then solves problems involving Fourier analysis of signals and generalized Fourier series. The problems cover topics like measuring performance of digital systems, classifying signals as energy or power, sketching signals, and approximating signals using generalized Fourier series.
Lecture 4 from virtual university of pakistanSaba Hanif
The document discusses error detecting and correcting techniques used in wireless networks. It reviews parity checks, cyclic redundancy checks (CRC), and block error correction codes. CRC uses a predetermined polynomial and modulo-2 arithmetic to generate a checksum over transmitted data blocks. This allows the receiver to detect errors by comparing the received checksum. Block error correction codes add redundancy to transmitted data blocks to allow the receiver to detect and correct a certain number of bit errors based on the code's Hamming distance.
High Speed Decoding of Non-Binary Irregular LDPC Codes Using GPUs (Paper)Enrique Monzo Solves
Implementation of a high speed decoding of non-binary irregular LDPC codes using CUDA GPUs.
Moritz Beermann, Enrique Monzó, Laurent Schmalen, Peter Vary
IEEE SiPS, Oct. 2013, Taipei, Taiwan
This document provides an overview of coding theory and recent advances in low-density parity-check (LDPC) codes. It discusses Shannon's channel coding theorem and how modern error-correcting codes achieve rates close to channel capacity. LDPC codes are described as having sparse parity-check matrices and being decoded iteratively using message passing. The performance of LDPC codes can be analyzed using density evolution and threshold calculations. Linear programming decoding is introduced as an alternative decoding approach that has connections to message passing decoding.
This document provides information about BCH codes, including:
1. BCH codes are linear cyclic block codes that can detect and correct errors. They allow flexibility in choosing block length and code rate.
2. Key characteristics of BCH codes include the block length being 2m - 1, error correction ability up to t errors where t<(2m - 1)/2, and minimum distance of at least 2t + 1.
3. Galois fields are finite fields that are important for constructing BCH codes. A generator polynomial is chosen based on the roots in the Galois field and is used to encode messages into codewords.
The document proposes a new normalization technique for compensating the optimistic output information produced by a soft output Viterbi algorithm (SOVA) decoder in a turbo decoder. The technique counts the sign differences between the a-priori and extrinsic information to determine a normalization factor for each data block. Simulations show the proposed technique achieves about a 0.2 dB coding gain improvement on average compared to other techniques, while reducing the number of iterations needed for decoding by up to 21.
This document discusses error detection and correction techniques at the data link layer. It covers different types of errors, the use of redundancy to detect or correct errors, block coding and convolutional coding approaches. Specific coding schemes like parity checks, cyclic redundancy checks (CRC), and Hamming codes are explained in detail. The key aspects covered are the use of redundant bits, minimum Hamming distance requirements for detection and correction capabilities, and how techniques like CRC and Hamming codes function to detect and correct single-bit errors. Assignments and example problems are also listed.
The document outlines Reed-Solomon error correction codes. It discusses how Reed-Solomon codes encode data using a generator polynomial to produce parity check symbols. The document then describes how Reed-Solomon codes can decode errors using syndrome calculation, error location polynomials, and finding the error positions and values through algorithms like Forney's method and Chien search. Reed-Solomon codes are widely used in applications like CDs, DVDs, wireless communications and digital television for their ability to efficiently correct both random and burst errors.
This document discusses error detection and correction techniques at the data link layer. It covers various types of errors that can occur and how redundancy is used to detect or correct them. Error correcting codes like block codes and convolutional codes are introduced. Specific coding schemes like parity checks, cyclic redundancy checks (CRC), and Hamming codes are explained in detail. The document provides examples of how these codes are implemented and their performance characteristics in terms of detecting and correcting single and burst errors. Standard polynomials used in CRC and properties of good polynomials are also discussed.
Convolutional codes are a type of error correcting code where each coded output block depends not only on the corresponding input block but also on previous input blocks. The encoder contains shift registers and logic gates. Convolutional codes are characterized by parameters k (input bits), n (output bits), and m (memory order). The distance properties and performance of convolutional codes depends on the constraint length L, which is a function of k and m. Convolutional codes can be represented using trees, trellises, state diagrams or polynomials to describe the encoding process.
This document discusses convolutional codes. It begins by defining a convolutional code as an error-correcting code that transforms each m-bit information symbol into an n-bit symbol, where the code rate is m/n. It then provides an example of a (2,1,8) convolutional coder with specific generator sequences. It includes diagrams of the coder circuit and trellis and discusses encoding an example message sequence. It also provides questions and answers demonstrating encoding and decoding operations using MATLAB.
The document describes experiments to generate and demodulate various digital modulation schemes using MATLAB, including:
- ASK modulation and demodulation using an envelope detector.
- BPSK modulation by changing the phase of a carrier signal and demodulation using correlation.
- FSK modulation by changing the frequency of a carrier signal and demodulation using correlation and subtraction.
- QPSK modulation using four phases to transmit two bits per symbol and gray encoding of the dibits.
The document describes experiments on generating and demodulating amplitude shift keying (ASK), phase shift keying (PSK), and frequency shift keying (FSK) signals using MATLAB.
For ASK, binary data is used to modulate a carrier signal by varying its amplitude. Demodulation recovers the data using an envelope detector. For PSK, binary data modulates the phase of a carrier signal. Demodulation correlates the signal with a reference carrier. For FSK, binary data determines the frequency of the carrier signal. Demodulation correlates the signal with two reference carriers and compares the results.
The MATLAB program for each modulation scheme generates test signals, plots the results, and recovers the data
Proof of Kraft Mc-Millan theorem - nguyen vu hungVu Hung Nguyen
This document provides a proof of the Kraft-McMillan theorem, which establishes a necessary condition for uniquely decodable codes. It begins by defining the Kraft inequality and proving that if a code C is uniquely decodable, the sum of 2 to the power of the negative codeword lengths must be less than or equal to 1. It then shows that if this condition is not met, the nth power of K(C) cannot grow linearly, proving the inequality holds. The document concludes by constructing a prefix code that satisfies a given set of codeword lengths meeting the Kraft inequality condition.
This document presents the design and implementation of an FPGA-based BCH decoder. It discusses BCH codes, which are binary error-correcting codes used in wireless communications. The implemented decoder is for a (15, 5, 3) BCH code, meaning it can correct up to 3 errors in a block of 15 bits. The decoder uses a serial input/output architecture and is implemented using VHDL on a FPGA device. It performs BCH decoding through syndrome calculation, running the Berlekamp-Massey algorithm to solve the key equation, and using Chien search to find error locations. The simulation result verifies correct decoding operation.
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
Full-RAG: A modern architecture for hyper-personalizationZilliz
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
TrustArc Webinar - 2024 Global Privacy SurveyTrustArc
How does your privacy program stack up against your peers? What challenges are privacy teams tackling and prioritizing in 2024?
In the fifth annual Global Privacy Benchmarks Survey, we asked over 1,800 global privacy professionals and business executives to share their perspectives on the current state of privacy inside and outside of their organizations. This year’s report focused on emerging areas of importance for privacy and compliance professionals, including considerations and implications of Artificial Intelligence (AI) technologies, building brand trust, and different approaches for achieving higher privacy competence scores.
See how organizational priorities and strategic approaches to data security and privacy are evolving around the globe.
This webinar will review:
- The top 10 privacy insights from the fifth annual Global Privacy Benchmarks Survey
- The top challenges for privacy leaders, practitioners, and organizations in 2024
- Key themes to consider in developing and maintaining your privacy program
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
UiPath Test Automation using UiPath Test Suite series, part 5DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 5. In this session, we will cover CI/CD with devops.
Topics covered:
CI/CD with in UiPath
End-to-end overview of CI/CD pipeline with Azure devops
Speaker:
Lyndsey Byblow, Test Suite Sales Engineer @ UiPath, Inc.
Introducing Milvus Lite: Easy-to-Install, Easy-to-Use vector database for you...Zilliz
Join us to introduce Milvus Lite, a vector database that can run on notebooks and laptops, share the same API with Milvus, and integrate with every popular GenAI framework. This webinar is perfect for developers seeking easy-to-use, well-integrated vector databases for their GenAI apps.
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
2. Timo O. Korhonen, HUT Communication Laboratory
2
Targets today
Taxonomy of coding
How cyclic codes are defined?
Systematic and nonsystematic codes
Why cyclic codes are used?
How their performance is defined?
How practical encoding and decoding circuits are realized?
How to construct cyclic codes?
3. Timo O. Korhonen, HUT Communication Laboratory
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Cryptography
(Ciphering)
Source
Coding
Compression
Coding
Line Coding
Error Control
Coding
Error
Correction
Coding
Error
Detection
Coding
- Secrecy/ Security
- Encryption (DES)
- Redundancy removal:
- Destructive (jpeg, mpeg)
- Non-destructive (zip)
- Makes bits
equal
probable
- Strives to
utilize
channel
capacity by
adding
extra bits
- for baseband
communications
- RX synchronization
- Spectral shaping
for BW requirements
- error detection
- used
in ARQ
as in TCP/IP
- feedback channel
- retransmissions
- quality paid by delay
= FEC
- no feedback
channel
- quality paid
by redundant
bits
Taxonomy
of
Coding
FEC: Forward Error Correction
ARQ: Automatic Repeat Request
DES: Data Encryption Standard
4. Timo O. Korhonen, HUT Communication Laboratory
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Background
Coding is used for
– error detection and/or error correction (channel coding)
– ciphering (security) and compression (source coding)
In coding extra bits are added or removed in data transmission
Channel coding can be realized by two approaches
– FEC (forward error coding)
block coding, often realized by cyclic coding
convolutional coding
– ARQ (automatic repeat request)
stop-and-wait
go-back-N
selective repeat … etc.
Note: ARQ applies FEC for error detection
5. Timo O. Korhonen, HUT Communication Laboratory
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Block and
convolutional coding
Block coding: mapping of source bits of length k into (binary) channel
input sequences n (>k) - realized by cyclic codes!
Binary coding produces 2k code words of length n. Extra bits in the code
words are used for error detection/correction
(1) block, and (2) convolutional codes:
– (n,k) block codes: Encoder output of
n bits depends only on the k input bits
– (n,k,L) convolutional codes:
each source bit influences n(L+1)
encoder output bits
– n(L+1) is the constraint length
– L is the memory depth
Essential difference of block and conv. coding
is in simplicity of design of encoding and decoding circuits
(n,k)
encoder
k bits n bits
k input bits
n output bits
n(L+1) output bits
input bit
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Why cyclic codes?
For practical applications rather large n and k must be used. This is
because in order to correct up to t errors it should be that
Hence for , large n and k
must be used (next slide)
Cyclic codes are
– linear: sum of any two code words is a code word
– cyclic: any cyclic shift of a code word produces another code word
Advantages: Encoding, decoding and syndrome computation easy by
shift registers
1
2 1 ...
1 2
n k
n
t
i
n
i
n n n
t
number of syndromes
(or check-bit error patterns)
number of error patters
in encoded word
2 1
1
1 log note: (1 )
C C
t
i
n
i
R q n k n R
n
/ 1
C
R k n
(n,k)
block coder
k-bits n-bits
7. Timo O. Korhonen, HUT Communication Laboratory
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Example
Consider a relatively high SNR channel such that only 1 or 2 bit errors
are likely to happen. Consider the ration
Take a constant code rate of Rc=k/n=0.8 and consider e with some
values of larger n and k :
This demonstrates that long codes are more advantages when a high
code rate and high error correction capability is required
(10,8) 0.35, (32,24) 0.89, (50,40) 0.97
e e e
(n,k)
block coder
k-bits n-bits
Number of 2-bit error patterns
Number of check-bits
/
C
R k n
2
1
2
1
1 log
( , )
log
1 2
t
c
i
n
R
i
n
n k
n k
n n
e
=
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Some block codes that can be realized by cyclic codes
(n,1) Repetition codes. High coding gain (minimum distance always n-
1), but very low rate: 1/n
(n,k) Hamming codes. Minimum distance always 3. Thus can detect 2
errors and correct one error. n=2m-1, k = n - m,
Maximum-length codes. For every integer there exists a
maximum length code (n,k) with n = 2k - 1,dmin = 2k-1.
BCH-codes. For every integer there exist a code with n = 2m-1,
and where t is the error correction capability
(n,k) Reed-Solomon (RS) codes. Works with k symbols that consists of
m bits that are encoded to yield code words of n symbols. For these
codes and
Nowadays BCH and RS are very popular due to large dmin, large number
of codes, and easy generation
Code selection criteria: number of codes, correlation properties, code
gain, code rate, error correction/detection properties
3
k
3
m
k n mt min
2 1
d t
2 1,number of check symbols 2
m
n n k t min
2 1
d t
1: Task: find out from literature what is meant by dual codes!
3
m
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Defining cyclic codes: code polynomial
and generator polynomial
An (n,k) linear code X is called a cyclic code when every cyclic shift of
a code X, as for instance X’, is also a code, e.g.
Each (n,k) cyclic code has the associated code vector with the n-bit code
polynomial
Note that the (n,k) code vector has the polynomial of degree of n-1 or
less. Mapping between code vector and code polynomial is one-to-one,
e.g. they specify each other uniquely
Manipulation of the associated polynomial is done in a Galois field (for
instance GF(2)) having elements {0,1}, where operations are performed
mod-2. Thus results are always {0,1} -> binary logic circuits applicable
For each cyclic code, there exists only one generator polynomial whose
degree equals the number of check bits q=n-k in the encoded word
1 2
1 2 1 0
( ) n n
n n
p x p x p x p x
X
1 2 1 0
( )
n n
x x x x
X 2 3 0 1
' ( )
n n n
x x x x
X
1 2
2 3 0 1
'( ) n n
n n n
p x p x p x p x
X
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Example: Generating of (7,4) cyclic code,
by generator polynomial G(p)=p3 +p+1
3 2
3
3 3 2 3 2 3 2
6 5 3
(1101) 1
(1011) 1
( 1 ) ( 1 ) 1
p p
p p
p p p p p p p p
p p p
M
G
X MG
4 3
p p
3 2
6 5 4 3 2
1
1 (1111111)
p p p
p p p p p p
<- message
<- encoded word
<- generator
The same result obtained by Maple:
11. Timo O. Korhonen, HUT Communication Laboratory
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Rotation of cyclic code yields another cyclic code
Theorem: A single cyclic shift of X is obtained by multiplication of pX
where after division by the factor pn+1 yields a cyclic code at the
remainder:
and by induction, any cyclic shift i is obtained by
Example:
Important point of implementation is is that the division by pn+1 can be
realized by a tapped shift register.
'( ) ( )mod( 1)
n
p p p p
X X
( ) ( )
( ) ( )mod( 1)
i i n
p p p p
X X
not a three-bit code (1010),
divide by the common factor
3 3
( )
1 011
1
1
1
p p p
p p
X
3
( )
p p p p
X
2
101 ( ) 1
p p
X
3 3
3
1
1
1
1
p p p
p
p
n-1 bit rotated code word
Shift left by 1 bit:
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Prove that
Note first that
then, by using (1) and (2)
Repeating the same division with higher degrees of p yields then
'( ) ( )mod( 1)
n
p p p p
X X
1 2
1 2 1 0
( ) n n
n n
p p x p x p x p x p
X
1
1 2
1 2 1 0
1 1
1 2
2 1 0 1
1 )
'( )
n
n n n
n n
n
n n
n
n n
x
p x p x p x p x p
x p x
x p x p x p x p
X
1 2
1 2 1 0
( ) n n
n n
p x p x p x p x
X
(1)
(2)
( ) ( )
( ) ( )mod( 1)
i i n
p p p p
X X
13. Timo O. Korhonen, HUT Communication Laboratory
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Cyclic codes and the common factor pn+1
Theorem: Cyclic code polynomial X can be generated by multiplying
the message polynomial M of degree k-1 by the generator polynomial G
of degree q=n-k where G is an q-th order factor of pn + 1.
Proof: assume message polynomial:
and the n-1 degree code is
or in terms of G
Consider then a shifted code version…
1 2
1 0
1 2
( ) k k
k k
p m p m p m p x
M
1 2
1 0
1 2
( ) ( ) ( ) ( ) ( )
k k
k k
p p m p p m p p m p p x
X MG G G G G
1 2
1 2 1 0
( ) n n
n n
p x p x p x p x
X
14. Timo O. Korhonen, HUT Communication Laboratory
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Now, if and assume G is a factor of pn+1 (not M),
then X’(p) must be a multiple of G that we actually already proved:
Therefore, X’ can be expressed by M1G for some other data vector M1
and X’ is must be a code polynomial.
Continuing this way for p(i)X(p), i = 2,3… we can see that X’’, X’’’ etc
are all code polynomial generated by the multiplication MG of the
respective, different message polynomials
Therefore, the (n,k) linear code X, generated by MG is indeed cyclic
when G is selected to be a factor of pn+1
2
2
1
1
1
1 2 1 0
1
2 1 0 1
( )
( 1) ( )
( 1) '( )
n
n
n
n
n n
n n
n
n n
p p x p x p x p x p
x p x p x p x p x
x p p p
X
X MG
( )
p p p
X MG
'( ) mod( 1)
n
X p p p
MG
G is a factor of pn+1
term has the factor pn+1 must be a multiple of G
15. Timo O. Korhonen, HUT Communication Laboratory
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Cyclic Codes & Common Factor
2
21 7 4 21, 3 7 21 1 , 7
1 3 3
2 3 2
3 3 1
x y p
x y
x y
x y
M
2
2
1
1
1
1 2 1 0
1
2 1 0 1
( )
( 1) ( )
( 1) '( )
n
n
n
n
n n
n n
n
n n
p p x p x p x p x p
x p x p x p x p x
x p p p
1
M G
X
X MG
16. Timo O. Korhonen, HUT Communication Laboratory
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Factoring cyclic code generator polynomial
Any factor of pn+1 with the degree of q=n-k
generates an (n,k) cyclic code
Example: Consider the polynomial p7+1. This can be factored as
Both the factors p3+p+1 or p3,+p2+1 can be used to generate an unique
cyclic code. For a message polynomial p2 +1 the following encoded
word is generated:
and the respective code vector (of degree n-1 or smaller) is
Hence, in this example
7 3 3 2
1 ( 1)( 1)( 1)
p p p p p p
2 3 5 2
( 1)( 1) 1
p p p p p p
(n,k)
cyclic encoder
k-bits n-bits
0101 0100111
0100111
3
7 4
q n k
n k
17. Timo O. Korhonen, HUT Communication Laboratory
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Example of Calculus of GF(2) in Maple
18. Timo O. Korhonen, HUT Communication Laboratory
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Encoder applies shift registers for multiplication of
data by the generator polynomial
Figure shows a shift register to realize multiplication by p3 +p+1
In practice, multiplication can be realized by two equivalent topologies:
unit delay
element
XOR-circuit
Data in
Encoded bits
x0
x1
xn-1
Note that the tap order
is opposite in these
topologies
Fibonacci-form
Galois-form Delay
element
19. Timo O. Korhonen, HUT Communication Laboratory
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Example: Multiplication of data by a shift register
out
1 1 0 0 0 0 0 0 0 0 0
0 1 1 0 0 0 0 0 0 0 0
0 0 1 1 0 0 0 0 0 0 0
0 0 0 1 1 0 0 0 0 0 1
0 0 0 0 1 1 0 0 0 0 1
0 0 0 0 0 1 1 0 0 0 1
0 0 0 0 0 0 1 1 0 0 0
0 0 0 0 0 0 0 1 1 0 1
0 0 0 0 0 0 0 0 1 1 0
generator polynomial
determines connection
of taps
word to be
encoded
Encoded word
3
4 2
( 1)( 1)
p p p
p p p
3
p p
4 3 2
1
1 11101
p p p
x0
x1
x3
1 2
1 2 1 0
( ) n n
n n
p x p x p x p x
X
20. Timo O. Korhonen, HUT Communication Laboratory
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Determines tap
connections
Word to be rotated
(divided by the common factor)
Adding the dashed-line (feedback)
enables division by pn+1
Remainder
Calculating the remainder (word rotation) by a shift
register
X A B C D
0 0 0 0 0 0 0 0
0 1 0 1 1 0 0 1
0 0 1 0 1 1 0 1
0 0 0 1 0 1 1 1
0 0 0 0 1 0 1 0
0 0 0 0 0 1 0 1
2
3
101 ( ) 1
( )
p p
p p p p
X
X
3 3
( )
1 011
1
1
1
p p p
p p
X
Remainder is left to the
shift register
1 0 1
load / read
Alternate way to
realize rotation
x0
xn-1
Maple script:
21. Timo O. Korhonen, HUT Communication Laboratory
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Examples of cyclic code generator polynomials
The generator polynomial for an (n,k) cyclic code is defined by
and G(p) is a factor of pn+1, as noted earlier. Any factor of pn+1 that has
the degree q (the number of check bits) may serve as the generator
polynomial. We noticed earlier that a cyclic code is generated by the
multiplication
where M(p) is the k-bit message to be encoded
Only few of the possible generating polynomials yield high quality
codes (in terms of their minimum Hamming distance)
1
1 1
( ) 1,
q q
q
p p p g pg q n k
G
( ) ( ) ( )
p p p
X M G
Some cyclic codes:
3
( ) 0 1
p p p
G
22. Timo O. Korhonen, HUT Communication Laboratory
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Systematic cyclic codes
Define the length q=n-k check vector C and the length-k message
vector M by
Thus the systematic n:th degree codeword polynomial is
1
1 1 0
( ) k
k
p m p m p m
M
1
1 1 0
( ) q
q
p c p c p c
C
1
1 1 0
1
1 1 0
( ) ( )
( ) ( )
n k k
k
q
q
q
p p m p m p m
c p c p c
p p p
X
M C
How to determine the check-bits??
Question: Why these denote the message bits still
the message bits are M(p) ???
check bits
message bits
(n,k)
cyclic encoder
k-bits n-bits
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Determining check-bits
Note that the check-vector polynomial is the remainder left over
after dividing
( ) ( ) ( ) ( ) ( )
q
p p p p p p
X M G M C
( ) ( )
( )
( ) ( )
n k
p p p
p
p p
M C
M
G G
( )/ ( )
n k
p p p
M G
( )
p
C
Example: (7,4) Cyclic code:
( ) mod ( )/ ( )
n k
p p p p
C M G
1010 -> 1010001
3 2
3
7 4 6 4
( ) 1
( )
( )
p p p
p p p
p p p p
G
M
M
3 2
3 3 6 4
3 2 3 2 6 4
( )
( )
( )/ ( ) 1 1
( ) ( ) ( ) 1 1
( ) ( ) ( 1)( 1) 1
n k
n k
p
p
p p p p p
p p p p p p p p
p p p p p p p p
C
Q
M G
M C
Q G
Definition of systematic cyclic code
7 7 5 1
1
5 5
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Division of the generated code by the generator
polynomial leaves no reminder
3 2
3 2 6 4
6 5 3
5 4 3
5 4 2
3 2
3 2
1
1 1
1
1
1
p p
p p p p
p p p
p p p
p p p
p p
p p
3 2
3 3 6 4
3 2 3 2 6 4
( )
( )
( )/ ( ) 1 1
( ) ( ) ( ) 1 1
( ) ( ) ( 1)( 1) 1
n k
n k
p
p
p p p p p
p p p p p p p p
p p p p p p p p
C
Q
M G
M C
Q G
3 2
3
7 4 6 4
( ) 1
( )
( )
p p p
p p p
p p p p
G
M
M
This can be used for error
detection/correction as we inspect later
25. Timo O. Korhonen, HUT Communication Laboratory
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Circuit for encoding systematic cyclic codes
We noticed earlier that cyclic codes can be generated by using shift
registers whose feedback coefficients are determined directly by the
generating polynomial
For cyclic codes the generator polynomial is of the form
In the circuit, first the message flows to the shift register, and feedback
switch is set to ‘1’, where after check-bit-switch is turned on, and the
feedback switch to ‘0’, enabling the check bits to be outputted
1
0
1 2
1 2 1
( ) 1
q q q
q q
p p p g p g pg
G
26. Timo O. Korhonen, HUT Communication Laboratory
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Decoding cyclic codes
Every valid, received code word R(p) must be a multiple of G(p),
otherwise an error has occurred. (Assume that the probability of noise to
convert code words to other code words is very small.)
Therefore dividing the R(p)/G(p) and considering the remainder as a
syndrome can reveal if an error has happed and sometimes also to reveal
in which bit (depending on code strength)
Division is accomplished by a shift registers
The error syndrome of q=n-k bits is therefore
This can be expressed also in terms of the error E(p) and the
code word X(p) while noting that the received word is in terms of error
( ) mod ( )/ ( )
p p p
S R G
( ) ( ) ( )
p p p
R X E
( ) mod ( ) ( ) / ( )
( ) mod ( )/ ( )
p p p p
p p p
S X E G
S E G
hence
27. Timo O. Korhonen, HUT Communication Laboratory
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Decoding cyclic codes: syndrome table
16.20 ( ) mod ( )/ ( )
s x e x g x
Using denotation of this example:
28. Timo O. Korhonen, HUT Communication Laboratory
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( )
g x
( ) mod ( )/ ( )
s x r x g x
Table 16.6
Decoding cyclic codes: error correction
29. Timo O. Korhonen, HUT Communication Laboratory
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Decoding circuit for (7,4) code
syndrome computation
To start with, the switch is at “0” position
Then shift register is stepped until all the received code bits have
entered the register
This results is a 3-bit syndrome (n - k = 3 ):
that is then left to the register
Then the switch is turned to the position “1” that drives the
syndrome out of the register
Note the tap order for Galois-form shift register
3
( ) 1
p p p
G
1
0
received code syndrome
x0 x1 xn-1
( ) mod ( )/ ( )
p p p
S R G
30. Timo O. Korhonen, HUT Communication Laboratory
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Lessons learned
You can construct cyclic codes starting from a given factored pn+1
polynomial by doing simple calculations in GF(2)
You can estimate strength of designed codes
You understand how to apply shift registers with cyclic codes
You can design encoder circuits for your cyclic codes
You understand how syndrome decoding works with cyclic codes and
you can construct the respect decoder circuit