12.1 Tanner Graphs and Code Construction
3 min read•august 7, 2024
Tanner graphs are a powerful tool for visualizing . They use variable and to represent bits and parity-check equations, with edges showing connections. Understanding these graphs is key to designing effective LDPC codes.
The is another way to define LDPC codes. It's a where rows and columns correspond to check and . Constructing good LDPC codes involves carefully crafting this matrix or its equivalent .
Graph Representation
Tanner Graph and its Components
Top images from around the web for Tanner Graph and its Components
Research of Multi-Rate LDPC Decoding in Optical Communication System View original
Is this image relevant?
Research of Multi-Rate LDPC Decoding in Optical Communication System View original
Is this image relevant?
1 of 1
Top images from around the web for Tanner Graph and its Components
Research of Multi-Rate LDPC Decoding in Optical Communication System View original
Is this image relevant?
Research of Multi-Rate LDPC Decoding in Optical Communication System View original
Is this image relevant?
1 of 1
- Tanner graph represents the parity-check matrix of an LDPC code using a structure
- Bipartite graph consists of two disjoint sets of nodes: variable nodes and check nodes
- Variable nodes correspond to the bits of the codeword
- Check nodes represent the parity-check equations that the codeword must satisfy
- Edges in the Tanner graph connect variable nodes to check nodes, indicating the participation of a bit in a specific parity-check equation
- of a Tanner graph is the length of the shortest cycle in the graph
- Larger girth generally leads to better error-correction performance (girth-12 )
Importance of Tanner Graph Properties
- Structure and connectivity of the Tanner graph significantly impact the performance of the LDPC code
- of variable and check nodes affects the code's ability to correct errors
- Degree distribution refers to the number of edges connected to each node
- Cycles in the Tanner graph can introduce dependencies among parity-check equations, potentially limiting the code's effectiveness
- Careful design of the Tanner graph is crucial for constructing high-performance LDPC codes (IEEE 802.11n Wi-Fi standard uses LDPC codes)
Parity-Check Matrix
Representation and Characteristics
- Parity-check matrix is a binary matrix that defines the parity-check equations of an LDPC code
- Each row of corresponds to a check node in the Tanner graph, and each column corresponds to a variable node
- Entry is 1 if variable node participates in the parity-check equation of check node , and 0 otherwise
- LDPC codes are characterized by a sparse parity-check matrix, meaning has a low density of 1's
- Sparsity enables efficient decoding algorithms ()
Relationship with Tanner Graph
- Parity-check matrix and Tanner graph are equivalent representations of an LDPC code
- Tanner graph can be derived from the parity-check matrix by creating a variable node for each column and a check node for each row
- Edges in the Tanner graph correspond to the non-zero entries in the parity-check matrix
- Properties of the parity-check matrix, such as row and column weights, translate to node degrees in the Tanner graph (regular and irregular LDPC codes)
Code Construction Methods
Random Construction
- Random construction involves generating a random sparse parity-check matrix that satisfies certain constraints
- Constraints may include a desired degree distribution for variable and check nodes
- Advantages of random construction include flexibility in code design and potential for good error-correction performance
- Drawbacks include lack of structure, which can make encoding and decoding more complex ()
Structured Construction
- Structured construction methods aim to introduce regularity and patterns into the parity-check matrix
- Common structured construction techniques include:
- Algebraic construction using finite fields and combinatorial designs ()
- Protograph-based construction, where a small base graph is replicated and permuted to form the full parity-check matrix
- Structured LDPC codes offer benefits such as simplified encoding and decoding, as well as improved error floor performance
Quasi-Cyclic LDPC Codes
- Quasi-cyclic (QC) LDPC codes are a class of structured LDPC codes with a particular form of the parity-check matrix
- Parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or zero matrices
- is obtained by cyclically shifting the rows of an identity matrix
- QC-LDPC codes have advantages in terms of encoding and decoding complexity, as well as hardware implementation (5G New Radio standard uses QC-LDPC codes)
- Proper design of the base matrix and permutation patterns is essential for achieving good performance with QC-LDPC codes
Key Terms to Review (17)
Bipartite graph: A bipartite graph is a type of graph where the set of vertices can be divided into two distinct sets such that no two vertices within the same set are connected by an edge. This structure is essential for modeling relationships in various applications, including coding theory, as it helps in constructing Tanner graphs for error-correcting codes. The properties of bipartite graphs make them suitable for representing relationships between two different types of entities, like variable nodes and check nodes in coding contexts.
Check Nodes: Check nodes are a fundamental component of Tanner graphs, which are used in the representation of error-correcting codes. In this context, check nodes serve as verification points that connect to variable nodes, ensuring that the codewords satisfy certain parity-check conditions. These nodes play a crucial role in decoding processes by allowing the detection and correction of errors that may occur during data transmission.
Circulant Permutation Matrix: A circulant permutation matrix is a special type of square matrix that represents a circular shift of elements in a vector. Each row of the matrix is a right cyclic shift of the row above it, making it a powerful tool in coding theory and linear algebra for efficiently representing permutations and transformations.
David J.C. MacKay: David J.C. MacKay was a renowned researcher and professor known for his significant contributions to the fields of coding theory and information theory. He is best recognized for his work on practical codes, particularly low-density parity-check (LDPC) codes, which are essential for efficient error correction in communication systems. MacKay's research emphasized the importance of combining theoretical foundations with practical applications, leading to advancements in how coding techniques are implemented in real-world scenarios.
Degree distribution: Degree distribution is a statistical measure that describes the number of edges connected to each vertex in a graph. In the context of coding theory, especially for low-density parity-check (LDPC) codes, understanding degree distribution helps in analyzing the performance and error correction capabilities of the code. It influences the construction of Tanner graphs and is essential for efficient encoding techniques used in LDPC codes.
Error Correction: Error correction is the process of detecting and correcting errors that occur during data transmission or storage. This method ensures the integrity and reliability of data by enabling systems to identify mistakes and recover the original information through various techniques.
Euclidean Geometry LDPC Codes: Euclidean Geometry LDPC Codes are a class of low-density parity-check codes derived from the principles of Euclidean geometry, characterized by their graphical representation through Tanner graphs. These codes utilize a sparse bipartite graph structure where variable nodes and check nodes interact in a way that allows for efficient error correction, making them particularly suitable for applications in data transmission and storage.
Girth: Girth, in the context of coding theory, specifically refers to the length of the shortest cycle in a graph. It plays a significant role in understanding the properties of Tanner graphs, which are used to represent low-density parity-check (LDPC) codes. A smaller girth in a Tanner graph can lead to poorer error correction performance, as it may indicate the presence of short cycles that can create correlation among variable nodes, affecting the decoding process.
LDPC Codes: Low-Density Parity-Check (LDPC) codes are a type of error-correcting code that allows for efficient communication over noisy channels. They are characterized by their sparse parity-check matrices, which lead to a simple structure that enables effective decoding algorithms. LDPC codes are particularly significant in modern coding techniques, known for their performance close to the Shannon limit, and play a crucial role in various applications, including digital communications and data storage.
Mackay-neal codes: Mackay-Neal codes are a type of error-correcting code that leverage the properties of bipartite graphs, particularly Tanner graphs, to achieve efficient decoding and reliable communication over noisy channels. These codes are specifically designed to facilitate the construction of low-density parity-check (LDPC) codes, which are known for their performance in correcting errors while maintaining a low computational complexity.
Parity-check matrix: A parity-check matrix is a mathematical representation used in coding theory that helps detect errors in transmitted messages by verifying the parity of codewords. It consists of a matrix where each row represents a linear equation that relates to the bits of a codeword, providing a way to check whether the received codeword is valid or not. This matrix plays a crucial role in error detection and correction techniques, influencing systematic encoding, syndrome decoding, and the construction of various codes.
Qc-ldpc codes: QC-LDPC codes, or Quasi-Cyclic Low-Density Parity-Check codes, are a class of error-correcting codes characterized by their structured construction using quasi-cyclic block matrices. These codes have gained attention for their ability to achieve near-capacity performance on various communication channels while maintaining efficient decoding algorithms. The quasi-cyclic nature of these codes allows for simplified encoding and decoding processes by exploiting the underlying structure, making them suitable for applications in modern digital communications.
Robert G. Gallager: Robert G. Gallager is a prominent figure in the field of coding theory, known for his contributions to error-correcting codes and the development of graphical models for representing these codes. His work, particularly on low-density parity-check (LDPC) codes, has been influential in enhancing data transmission reliability and efficiency. His concepts have paved the way for innovative approaches in coding theory, especially through the use of Tanner graphs for code construction.
Sparse binary matrix: A sparse binary matrix is a two-dimensional array where most of the elements are zero, and the remaining elements are typically either one or zero. This type of matrix is particularly important in coding theory as it is used to represent the connections between variable nodes and check nodes in Tanner graphs, facilitating efficient encoding and decoding processes for error correction codes.
Sum-product algorithm: The sum-product algorithm is a message-passing algorithm used for inference in graphical models, particularly in the context of decoding error-correcting codes. It operates on factor graphs or Tanner graphs by passing messages between variable nodes and check nodes, facilitating efficient computation of marginal distributions. This algorithm plays a critical role in decoding processes and is foundational for belief propagation techniques, enabling iterative decoding of codes while balancing complexity and performance.
Tanner Graph: A Tanner graph is a bipartite graph used to represent the relationships between variables and constraints in coding theory, particularly for linear block codes and low-density parity-check (LDPC) codes. It consists of two types of nodes: variable nodes, which represent the bits of the codeword, and check nodes, which represent the parity-check equations. This visual representation aids in understanding the structure of the code and facilitates efficient decoding algorithms.
Variable Nodes: Variable nodes are fundamental components in Tanner graphs that represent the bits of data in a coding scheme. Each variable node corresponds to a specific bit in the code and connects to check nodes, which represent constraints or parity-check equations related to those bits. These connections form a bipartite graph structure, crucial for understanding how information is processed and transmitted in error-correcting codes.