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Capacity-achieving codes are a class of error-correcting codes that can theoretically approach the maximum possible efficiency for data transmission over a noisy communication channel. The term "capacity" refers to the channel capacity, which is the maximum rate at which information can be transmitted over a communication channel with an arbitrarily low probability of error, as defined by Shannon's channel capacity theorem.

Capacity-approaching codes are a class of error-correcting codes that are designed to achieve performance close to the theoretical limits of capacity defined by Shannon's channel capacity theorem. Shannon's theorem states that there is a maximum rate of information that can be transmitted over a communication channel without error, given a particular signal-to-noise ratio. The challenge in practical communication systems is to approach this limit in a way that allows for reliable communication despite the presence of noise and other impairments.

A hash function is a mathematical algorithm that transforms input data (often called a message) into a fixed-size string of characters, which is typically a sequence of numbers and letters. This output is known as a hash value or hash code. Hash functions are widely used in various fields such as computer science, cryptography, and data integrity verification. ### Key Properties of Hash Functions: 1. **Deterministic**: For a given input, a hash function will always produce the same hash value.

Message Authentication Codes (MACs) are cryptographic constructs used to verify the integrity and authenticity of a message. A MAC is generated by applying a cryptographic hash function or a symmetric key algorithm to the message data combined with a secret key. This results in a fixed-size string of bits (the MAC), which is then sent along with the message. ### Key Features of MACs: 1. **Integrity**: MACs ensure that the message has not been altered in transit.

AN codes, also known as AN (Aerospace and National) codes, are a system of designations used to identify specific types of military and aerospace components, hardware, and materials. These codes are typically employed to standardize parts for use in aerospace applications, including various types of aircraft, spacecraft, and military vehicles. The AN designation system includes various categories, such as: 1. **AN Drones and Components**: Identifies parts specific to drones and unmanned vehicles.

In the context of data networks, "acknowledgment" (often abbreviated as "ACK") refers to a signal or message sent from a receiver to a sender to confirm the successful receipt of data. Acknowledgments play a crucial role in various network communication protocols, particularly in ensuring data integrity and reliability.

An **alternant code** is a type of linear error-correcting code that is particularly used in coding theory. Alternant codes are a subclass of algebraic codes that are constructed using properties of polynomial evaluations and are designed to correct multiple symbol errors.

Automated quality control of meteorological observations refers to the processes and systems used to ensure the accuracy, consistency, and reliability of data collected from weather stations and other meteorological instruments. Given the vast amount of data generated by these observations, automation helps in efficiently identifying and correcting data errors without the need for extensive manual intervention.

Automatic Repeat reQuest (ARQ) is an error control method used in data communication protocols to ensure the reliable transmission of data over noisy communication channels. The basic idea behind ARQ is to detect errors in transmitted messages and to automatically request the retransmission of corrupted or lost data packets.

BCH (Bose–Chaudhuri–Hocquenghem) codes are a class of error-correcting codes that are used in digital communication and storage to detect and correct multiple random error patterns in data. These codes are named after the three researchers who developed them in the 1960s: Raj Chandra Bose, Alexis Hocquenghem, and D. R. McEliece, who contributed to their understanding and application.

The BCJR algorithm, named after its authors Bahl, Cocke, Jelinek, and Raviv, is a well-known algorithm used for decoding convolutional codes, which are widely used in communication systems for error correction. The algorithm operates in the context of maximum a posteriori (MAP) estimation, enabling it to efficiently decode received signals by computing the most likely sequence of transmitted information bits based on the observed noisy signals.

Berger code is a method used in computer science and data encoding, specifically in the context of information theory and coding theory. It is a type of code used for the efficient representation of data in a way that minimizes the number of bits required to represent information, particularly for certain types of data structures like trees or binary data. The basic idea behind Berger coding is to create a unique encoding for each possible configuration of a dataset, allowing for efficient storage and retrieval of information.

The Berlekamp–Massey algorithm is a fundamental algorithm in coding theory and information theory used to find the shortest linear feedback shift register (LFSR) that can generate a given finite sequence of output. It is particularly useful for determining the linear recurrence relations for a sequence, which is essential in applications such as error correction coding, cryptography, and sequence analysis.

The Berlekamp–Welch algorithm is a mathematical algorithm used for error correction in coding theory, particularly in the context of Reed-Solomon codes. It is designed to efficiently decode received polynomial data that may have been corrupted by errors during transmission.

The Binary Golay code refers to a specific error-correcting code known as the Golay code, which is used in digital communications to protect data against errors during transmission or storage. There are two main types of Golay codes: the (23, 12, 7) binary Golay code and the (24, 12, 8) extended binary Golay code.

Binary Reed-Solomon encoding is a type of error-correcting code that is used to detect and correct errors in data storage and transmission. Reed-Solomon codes are based on algebraic constructs over finite fields, and the binary variant specifically deals with binary data (0s and 1s). ### Key Features of Binary Reed-Solomon Encoding: 1. **Error Correction**: Reed-Solomon codes can correct multiple bit errors in a block of data.

Bipolar violation typically refers to a situation in electrical engineering and telecommunications where a signal that is expected to alternate between two distinct states (commonly represented as positive and negative, or high and low) fails to do so appropriately. This can occur in systems using bipolar encoding, where the signal is represented using both positive and negative voltages. In a correctly functioning bipolar system, the signal should alternate between positive and negative voltages in a way that maintains an even distribution of these states.

Burst error-correcting codes are specialized error-correction codes designed to detect and correct a series of consecutive bits that have been corrupted in a communication channel. Unlike random errors, where a few bits might change here and there, burst errors involve a contiguous sequence of errors caused by factors like noise, interference, or signal degradation in transmission mediums.

Casting out nines is a mathematical technique used primarily for error detection in arithmetic calculations, especially addition and multiplication. The method relies on the concept of modular arithmetic, specifically modulo 9. The basic idea is to reduce numbers into a single-digit form called a "digit sum" or "reduced digit" by repeatedly adding the digits of a number until a single digit is obtained. This final digit, known as the "digital root," can be used to verify calculations.

A check digit is a form of redundancy check used for error detection on identification numbers, such as product codes, account numbers, and various types of identification numbers. It is a single digit added to the end of a number (or sometimes inserted at a specific position) that is calculated based on the other digits in that number. The purpose of the check digit is to help verify that the number has been entered or transmitted correctly.