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Graph theory is a branch of mathematics that studies the properties and applications of graphs, which are structures made up of nodes (vertices) connected by edges. It is fundamental in computer science, network analysis, and combinatorics for solving problems related to connectivity, flow, and optimization.
Low-density parity-check (LDPC) codes are a class of linear error-correcting codes that achieve near-optimal performance close to the Shannon limit, making them highly efficient for data transmission over noisy channels. They utilize sparse bipartite graphs for encoding and decoding, allowing for iterative algorithms that significantly reduce computational complexity.
Belief propagation is an algorithm used for performing inference on graphical models, such as Bayesian networks and Markov random fields, by iteratively updating and passing messages between nodes. It is particularly effective for computing marginal distributions and finding the most probable configurations in tree-structured graphs, but can also be applied to loopy graphs with approximate results.
Channel capacity is the maximum rate at which information can be reliably transmitted over a communication channel, as defined by Shannon's noisy channel coding theorem. It represents the upper bound of data transmission efficiency, taking into account noise and interference in the channel.
Sparse matrices are matrices predominantly filled with zero elements, making them suitable for specialized storage techniques that save memory and computational resources. Efficient algorithms for Sparse matrices take advantage of their structure to perform operations like matrix multiplication and inversion without processing the zero elements.
Turbo Codes are a class of high-performance error correction codes that approach the Shannon limit, enabling reliable data transmission over noisy channels. They use iterative decoding and a combination of two or more convolutional codes, significantly improving the error correction capability compared to traditional methods.
Information theory is a mathematical framework for quantifying information, primarily focusing on data compression and transmission efficiency. It introduces fundamental concepts such as entropy, which measures the uncertainty in a set of outcomes, and channel capacity, which defines the maximum rate of reliable communication over a noisy channel.
Shannon's Theorem, also known as the Shannon-Hartley theorem, establishes the maximum data rate or channel capacity that can be achieved over a communication channel with a specified bandwidth and signal-to-noise ratio, without error. It forms the foundation of information theory by quantifying the trade-off between bandwidth, noise, and data transmission rate, guiding the design of efficient communication systems.
Gallager codes, also known as Low-Density Parity-Check (LDPC) codes, are a class of linear error-correcting codes that are defined by sparse bipartite graphs. They are known for their capacity-approaching performance on noisy channels and efficient decoding algorithms, making them highly effective in modern communication systems.
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