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Kodning och avkodning av Reed-Muller koder

Meyer, Linda

Published Jan 1, 2021• 0 citations
OpenAlex

Abstract

In this thesis some families of linear error correcting codes are presented. The reader will find a general description of binary codes and more specific details about linear codes such as Hamming, repetition codes, Reed-Muller codes, etc. To fully immerse ourselves in the methods of coding and decoding, we will introduce examples in order to contribute to the understanding of the theories. In these times of much communication through computer technology, our daily lives involve a substantial amount of data transmission. It is essential that these data are transmitted without errors through the communication channels. Therefore, the scientific field of error-correcting codes holds a significant importance in many aspects of todays society. The main goal of this thesis is to study linear block codes which belong to the class of binary codes. In this case we will attribute a more prominent role to first order Reed-Muller codes.

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量子 Reed-Müller 碼

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In this thesis, we start from the theory of classical error-correcting codes. We also introduce classical Reed-Muller codes along with its properties. We discuss the encoding and decoding methods for classical Reed-Muller codes using the majority rule. Then we turn to the topic of quantum error-correcting codes.We talk about stabilizer codes and discuss how classical Reed-Muller codes can be transformed into quantum Reed-Muller codes using Steane's method. Finally, we give a construction of a quantum code with parameter [[8,3,3]], and discuss its encoding and decoding process.

•Jan 1, 2011•0 citations

Reed-Muller Codes: Spherically-Punctured Codes and Decoding Algorithms

Olga Kapralova

Reed-Muller (RM) codes are classical codes that have enjoyed unabated interest since their introduction in 1954 due to their simple recursive structure and low-complexity decoding. The main goal of the thesis is to develop new coding techniques that can improve the performance of RM codes. In this work, we wish to keep the recursive low-complexity structure of RM codes while extending the sets of possible parameters and decoding algorithms. We address this task using two different methods. First, we consider new code constructions that extend code parameters using puncturing. Second, we design new decoding algorithms that repeatedly use the gradient properties of the classic recursive structure to improve code performance of RM codes. In the first part of the thesis, we propose a new class of codes that enjoy the recursive structure of RM codes, but reduce the exponential length of RM codes to a polynomial in m. These codes are obtained by puncturing classical RM codes, i.e. by restricting binary RM codes to small subsets of their positions. We study new recursive properties of these codes and analyze their parameters. In the second part of the thesis we develop new decoding algorithms for binary RM codes of second order that achieve near optimal performance for short and moderate code lengths. RM codes enjoy some simple decoding procedures; however, these algorithms fall short of optimal performance on moderate block lengths. We propose a suboptimal algorithm that reduces the complexity of maximum-likelihood decoding from super-polynomial order O(n(m+1)/2) to polynomial order O(mn2). At the same time, our numerical simulations show that the new algorithm performs very closely to its maximum-likelihood counterpart for blocks of up to several thousand bits.

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A survey of reed-muller codes from polar coding perspective

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IEEE Information Theory Workshop (ITW)•Jan 1, 2010•58 citations

Recursive Decoding and Its Performance for Low-Rate Reed-Muller Codes

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Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length $n$ and fixed order $r.$ An algorithm is designed that has complexity of order $n\log n$ and corrects most error patterns of weight up to $n(1/2-\varepsilon)$ given that $\varepsilon$ exceeds $n^{-1/2^{r}}.$ This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.

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Reed-Muller Codes Achieve Capacity on BMS Channels

Galen Reeves, Henry D. Pfister

This paper considers the performance of long Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori decoding. Its main result is that the family of binary RM codes achieves capacity on any BMS channel with respect to bit-error rate. This resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

•Oct 27, 2021•16 citations
In this thesis some families of linear error correcting codes are presented. The reader will find a general description of binary codes and more specific details about linear codes such as Hamming, repetition codes, Reed-Muller codes, etc. To fully immerse ourselves in the methods of coding and decoding, we will introduce examples in order to contribute to the understanding of the theories. In these times of much communication through computer technology, our daily lives involve a substantial amount of data transmission. It is essential that these data are transmitted without errors through the communication channels. Therefore, the scientific field of error-correcting codes holds a significant importance in many aspects of todays society. The main goal of this thesis is to study linear block codes which belong to the class of binary codes. In this case we will attribute a more prominent role to first order Reed-Muller codes.
In this thesis, we start from the theory of classical error-correcting codes. We also introduce classical Reed-Muller codes along with its properties. We discuss the encoding and decoding methods for classical Reed-Muller codes using the majority rule. Then we turn to the topic of quantum error-correcting codes.We talk about stabilizer codes and discuss how classical Reed-Muller codes can be transformed into quantum Reed-Muller codes using Steane's method. Finally, we give a construction of a quantum code with parameter [[8,3,3]], and discuss its encoding and decoding process.
Reed-Muller (RM) codes are classical codes that have enjoyed unabated interest since their introduction in 1954 due to their simple recursive structure and low-complexity decoding. The main goal of the thesis is to develop new coding techniques that can improve the performance of RM codes. In this work, we wish to keep the recursive low-complexity structure of RM codes while extending the sets of possible parameters and decoding algorithms. We address this task using two different methods. First, we consider new code constructions that extend code parameters using puncturing. Second, we design new decoding algorithms that repeatedly use the gradient properties of the classic recursive structure to improve code performance of RM codes. In the first part of the thesis, we propose a new class of codes that enjoy the recursive structure of RM codes, but reduce the exponential length of RM codes to a polynomial in m. These codes are obtained by puncturing classical RM codes, i.e. by restricting binary RM codes to small subsets of their positions. We study new recursive properties of these codes and analyze their parameters. In the second part of the thesis we develop new decoding algorithms for binary RM codes of second order that achieve near optimal performance for short and moderate code lengths. RM codes enjoy some simple decoding procedures; however, these algorithms fall short of optimal performance on moderate block lengths. We propose a suboptimal algorithm that reduces the complexity of maximum-likelihood decoding from super-polynomial order O(n(m+1)/2) to polynomial order O(mn2). At the same time, our numerical simulations show that the new algorithm performs very closely to its maximum-likelihood counterpart for blocks of up to several thousand bits.
Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length nnn and fixed order r.r.r. An algorithm is designed that has complexity of order nlog⁡nn\log nnlogn and corrects most error patterns of weight up to n(1/2−ε)n(1/2-\varepsilon)n(1/2−ε) given that ε\varepsilonε exceeds n−1/2r.n^{-1/2^{r}}.n−1/2r. This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.
Error correcting codes is an area of mathematics that combines many theoretical concepts with a wide variety of applications.In this expository thesis, we will begin by introducing many of the fundamental ideas in classical error correcting codes.The goal of this is to introduce the reader to the field of coding theory and discuss a few of the wide range of classical codes that have been studied.The second section will discuss the recent findings in error correcting codes, namely their use in quantum error correcting codes.This section uses much more theoretical areas of math, when compared to the classical section, in the construction of quantum error correcting codes.Most of the research discussed in this section has been completed within the past six years.All of the research discoveries presented in this thesis have been previously known.
This paper considers the performance of long Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori decoding. Its main result is that the family of binary RM codes achieves capacity on any BMS channel with respect to bit-error rate. This resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.