Glossary

10.1 Linear block codes

2 min read•july 25, 2024

are essential for in digital communication. They use fixed-length inputs and outputs, with codewords containing information and redundancy bits. The measures efficiency, while the determines .

Encoding involves multiplying message vectors by a , while decoding uses for . The enables error correction. These codes can detect and correct multiple errors, with achieving optimal performance within theoretical limits.

Linear Block Codes Fundamentals

Properties of linear block codes

Top images from around the web for Properties of linear block codes

  • Hamming code - Wikipedia View original

    Is this image relevant?

  • coding theory - Hamming code error correction - Mathematics Stack Exchange View original

    Is this image relevant?

  • Hamming code - Wikipedia View original

    Is this image relevant?

1 of 3

Top images from around the web for Properties of linear block codes

  • Hamming code - Wikipedia View original

    Is this image relevant?

  • coding theory - Hamming code error correction - Mathematics Stack Exchange View original

    Is this image relevant?

  • Hamming code - Wikipedia View original

    Is this image relevant?

1 of 3
  • Linear block codes enhance error correction in digital communication systems
  • Linearity ensures sum of codewords yields another valid codeword
  • Block structure maintains fixed-length input and output
  • Codewords comprise nn bits total, kk information bits, r=n−kr = n - k redundancy bits
  • Code rate R=k/nR = k/n measures efficiency
  • Minimum Hamming distance dmind_{min} determines error-correcting capability
  • preserves information bits, appends parity bits

Matrices for linear block codes

  • Generator matrix G (k×nk \times n) facilitates message encoding
  • Systematic form G = [I_k | P] includes identity matrix I_k and parity submatrix P
  • Parity-check matrix H ((n−k)×n(n-k) \times n) enables error detection and correction
  • Systematic form H = [-P^T | I_{n-k}] incorporates transposed parity submatrix
  • G and H relationship: GHT=0GH^T = 0 (zero matrix)
  • utilizes H as its generator matrix

Encoding and Decoding

Encoding and decoding with linear block codes

  • Encoding multiplies message vector m by generator matrix G: c=mGc = mG
  • calculates s=rHTs = rH^T to detect errors
  • indicates error-free transmission
  • Non-zero syndrome reveals errors
  • Error correction identifies error pattern using syndrome
  • Standard array decoding organizes received vectors, uses coset leaders

Error capabilities of linear block codes

  • Error detection handles up to dmin−1d_{min} - 1 errors
  • Error correction manages up to ⌊(dmin−1)/2⌋\lfloor (d_{min} - 1) / 2 \rfloor errors
  • limits codewords for given n and dmind_{min}: 2k≤2n∑i=0t(ni)2^k \leq \frac{2^n}{\sum_{i=0}^{t} \binom{n}{i}}
  • Perfect codes (Hamming codes, Golay codes) meet Hamming bound equality
  • covers errors up to length r
  • correspond to non-zero codewords

Key Terms to Review (19)

Burst error detection: Burst error detection refers to the technique used in data transmission systems to identify and correct errors that occur in clusters or bursts, rather than as isolated single-bit errors. This concept is crucial for maintaining data integrity and reliability in communications, especially in systems utilizing linear block codes that enable error detection and correction through redundancy.
Code rate: Code rate is a measure that represents the efficiency of a coding scheme, defined as the ratio of the number of information bits to the total number of bits in the encoded message. A higher code rate indicates a more efficient code, as it means fewer redundant bits are added for error correction. Code rate plays a crucial role in determining the performance and reliability of different coding techniques, influencing trade-offs between error correction capability and data transmission efficiency.
Dual Code: A dual code is a type of error-correcting code that is derived from another linear code, typically referred to as the primal code. In this context, the dual code consists of all vectors that are orthogonal to the codewords of the primal code, providing a powerful way to analyze and compare the properties of linear codes. Understanding dual codes is essential for grasping how different codes can be utilized for efficient encoding and decoding processes.
Error Correction: Error correction is a set of techniques used to detect and correct errors in data transmission or storage. It ensures that the original information is accurately retrieved, even if errors occur during the process. This concept is crucial in maintaining the integrity of data across various modern technologies, such as communication systems and digital storage devices, where noise and interference can introduce inaccuracies.
Error detection: Error detection is the process of identifying errors in data transmission or storage, ensuring the integrity and reliability of the information being communicated. It involves various techniques that add redundancy to the transmitted data, allowing the receiver to check for discrepancies. Effective error detection mechanisms play a vital role in maintaining communication quality and minimizing data loss, especially in systems where accurate information transfer is critical.
Error-correcting capability: Error-correcting capability refers to the ability of a code to detect and correct errors that occur during the transmission or storage of data. This characteristic is crucial in ensuring data integrity, as it determines how many errors can be accurately identified and corrected without needing retransmission. In the context of coding theory, particularly with linear block codes, this capability is influenced by factors such as the code's minimum distance and redundancy.
Generator matrix: A generator matrix is a mathematical representation used in coding theory to create linear block codes. It generates the codewords of a linear block code from a set of input messages by performing matrix multiplication. This concept is crucial for understanding how data is encoded and ensures the integrity of information during transmission.
Golay Code: Golay code is a type of error-correcting code used in digital communications and data storage that can correct multiple errors in a block of data. It is specifically known for its ability to correct up to three errors in a 23-bit codeword, making it highly efficient for reliable data transmission. This code is classified as a linear block code and plays a significant role in ensuring data integrity and error correction in various applications.
Hamming Bound: The Hamming Bound is a crucial concept in coding theory that provides a limit on the maximum number of codewords that can be packed into a given space without overlap. This bound is important for understanding the error-correcting capabilities of linear block codes, as it helps to determine how many errors can be corrected based on the design of the code and its length. In essence, it ensures that codes can efficiently represent information while being robust against errors during transmission.
Hamming Code: Hamming Code is a set of error-correcting codes used to detect and correct single-bit errors in data transmission and storage. Named after Richard Hamming, it uses parity bits added to the data bits to create a code word, ensuring that errors can be identified and corrected efficiently. This technique is essential for maintaining data integrity in communication systems and computer memory.
Linear block codes: Linear block codes are a type of error-correcting code that transform a message of a fixed length into a codeword of a larger fixed length using linear combinations of the message symbols. These codes are structured in such a way that they can efficiently detect and correct errors that occur during data transmission, making them essential for reliable communication in digital systems.
Minimum Hamming Distance: Minimum Hamming distance is defined as the smallest number of positions in which two codewords differ within a given coding scheme. It is a critical metric for evaluating the error-detection and error-correction capabilities of linear block codes, as a higher minimum Hamming distance indicates a greater ability to identify and correct errors in transmitted data. This measure directly influences the code's effectiveness in maintaining data integrity during transmission over noisy channels.
Parity-check matrix: A parity-check matrix is a matrix used in coding theory that helps to identify errors in linear block codes by establishing a relationship between the transmitted codewords and the valid codewords. It serves as a tool for error detection, allowing one to determine whether a received message contains errors based on the linear equations derived from the matrix. By using the parity-check matrix, one can compute the syndrome of a received vector, which reveals information about the presence and location of any errors.
Perfect Codes: Perfect codes are a special class of error-correcting codes that can correct a specific number of errors while ensuring that the total number of codewords fills the entire space defined by the message length. They achieve maximum efficiency in terms of error correction, meaning there is no wasted space in the code. This makes them particularly valuable in applications where reliable data transmission is critical.
Syndrome calculation: Syndrome calculation is a method used in error detection and correction that helps identify the error pattern in received data. It involves using a parity-check matrix to calculate a syndrome vector, which indicates whether an error has occurred and, if so, its specific location. This process is vital for maintaining data integrity in linear block codes, allowing the decoder to pinpoint and correct errors efficiently.
Syndrome decoding: Syndrome decoding is a technique used in error detection and correction that leverages the concept of a 'syndrome' to identify and correct errors in transmitted codewords. It involves calculating a syndrome vector based on the received vector and comparing it to a predefined table or set of syndromes associated with potential error patterns. This method is particularly effective in linear block codes and cyclic codes, allowing for efficient error correction without needing to search through all possible codewords.
Systematic form: Systematic form refers to a structured representation of linear block codes where the message bits are positioned in a specific way, making it easier to encode and decode information. This form clearly distinguishes between the data and the parity bits, allowing for a more straightforward interpretation of the code structure. It enables efficient encoding and decoding processes, facilitating error detection and correction.
Undetectable errors: Undetectable errors are discrepancies that occur in transmitted data but go unnoticed by the error detection mechanism in place. These errors can lead to significant problems, especially in systems where data integrity is critical, as they can propagate without triggering any alerts or corrective actions. Understanding undetectable errors is essential when evaluating the reliability of error detection methods, particularly in linear block codes.
Zero Syndrome: Zero syndrome is a condition observed in linear block codes where the codeword corresponding to the all-zero input results in an all-zero output. This phenomenon is important because it indicates that the encoding scheme must handle zero inputs effectively, ensuring that the presence of a zero does not lead to ambiguous or misleading decoding. Understanding zero syndrome is essential for designing efficient error detection and correction mechanisms in communication systems.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide