Error Correction Codes

Contents

Recall the following terms: i. Block Codes ii. Linear Block Codes iii. Hamming Distance & Hamming Weight iv. Minimum Distance & Error Detection v. Correction Capabilities

i. Block codes: Block codes are a type of error correction code that operate on blocks of symbols, rather than on individual symbols. They are used to detect and correct errors in transmitted messages or signals by adding redundancy to the transmitted data.

ii. Linear block codes: Linear block codes are a type of block code that can be represented as a linear combination of a set of codewords. They are characterized by the property that the sum of any two codewords is also a valid codeword. Linear block codes are widely used in communication systems due to their simple structure and good error correction capabilities.

iii. a) Hamming distance: The Hamming distance between two codewords is the number of bit positions in which the two codewords differ. It is a measure of the similarity between two codewords and is an important concept in error correction coding.
b) Hamming weight: The Hamming weight of a codeword is the number of 1’s in the codeword. It is a measure of the complexity of the codeword and is often used as a measure of the error correction capabilities of a code.

iv. a) Minimum distance: The minimum distance of a code is the smallest Hamming distance between any two codewords in the code. It is an important measure of the error correction capabilities of the code, as it determines the number of errors that can be corrected by the code.
b) Error detection: Error detection is the process of detecting errors in transmitted messages or signals. It is an important aspect of communication systems, as it allows the receiver to identify and correct errors in the received data.

v. Correction capabilities: The correction capabilities of a code refer to the number of errors that the code can correct. A code with high correction capabilities is able to correct a large number of errors, while a code with low correction capabilities is only able to correct a small number of errors. The correction capabilities of a code are determined by its minimum distance and its error detection capabilities.

Relate the minimum Hamming Distance and Error detecting and Correcting Capabilities

The minimum Hamming distance of a code is an important measure of the error correction capabilities of the code. It determines the number of errors that the code can correct and the number of errors that it can detect.

For a given code, the minimum Hamming distance d is related to the error detection and correction capabilities of the code according to the following:

· The code can detect any number of errors up to d-1.

· The code can correct t or fewer errors, where t is equal to floor((d-1)/2).

For example, consider a code with a minimum Hamming distance of 3. This code can detect any number of errors up to 2 and can correct 1 or fewer errors.

On the other hand, consider a code with a minimum Hamming distance of 4. This code can detect any number of errors up to 3 and can correct 2 or fewer errors.

In general, a code with a larger minimum Hamming distance has higher error correction capabilities, as it is able to correct a larger number of errors. However, the minimum Hamming distance of a code also determines the overhead associated with the code, as more bits are required to represent the codewords with a larger minimum Hamming distance. As a result, there is a trade-off between the error correction capabilities of a code and the overhead associated with the code.

Recall the Generator Matrix and Parity – Check Matrix

The generator matrix and parity-check matrix are two important concepts in the design and analysis of error correction codes. They are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals.

A generator matrix is a matrix that defines the set of codewords in a code. It is used to encode messages or signals by multiplying the message by the generator matrix to produce the codeword.

A parity-check matrix is a matrix that is used to check the parity of a received codeword. It is used to decode messages or signals by multiplying the received codeword by the parity-check matrix and checking the resulting syndrome.

Both the generator matrix and the parity-check matrix are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals. They are important tools in the design and analysis of error correction codes and are widely used in communication systems.

Describe Encoding and Decoding of Linear Block Codes

Encoding and decoding of linear block codes are the processes of converting a message or signal into a codeword for transmission over a communication channel and recovering the original message or signal from the received codeword, respectively.

Encoding of linear block codes is typically performed using a generator matrix G. The generator matrix is a matrix that defines the set of codewords in the code. To encode a message or signal, the message is first represented as a vector m, and the codeword c is obtained by multiplying the message vector by the generator matrix:

c = mG

where c is the codeword, m is the message vector, and G is the generator matrix.

Decoding of linear block codes is typically performed using a parity-check matrix H. The parity-check matrix is a matrix that is used to check the parity of a received codeword. To decode a received codeword r, the syndrome s is obtained by multiplying the received codeword by the parity-check matrix:

s = rH

where s is the syndrome and H is the parity-check matrix.

If the syndrome is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the syndrome to identify and correct the errors in the received codeword. If the syndrome is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Encoding and decoding of linear block codes are important processes in communication systems, as they allow messages or signals to be transmitted over a communication channel with error correction. Linear block codes are widely used in communication systems due to their simple structure and good error correction capabilities.

Recall Hamming Bound and Hamming Code

The Hamming bound is a fundamental result in error correction coding that relates the minimum distance d of a code to its error correction and error detection capabilities. It states that the minimum number of codewords in a code with minimum distance d is given by:

2(n-k) ≤ 2(d-1)

where n is the length of the codewords, k is the number of information bits in the codewords, and d is the minimum distance of the code.

The Hamming bound is an important result in error correction coding, as it provides a limit on the minimum number of codewords in a code with a given minimum distance. It shows that the minimum number of codewords in a code increases with the minimum distance of the code, and that the minimum distance of a code determines the error correction and error detection capabilities of the code.

Hamming code is a specific type of error correction code that was developed by Richard Hamming. It is a linear block code that is based on the idea of adding redundant bits to a message or signal to detect and correct errors. Hamming codes are characterized by the property that the minimum distance of the code is 3, which allows them to correct a single error and detect any two errors. They are widely used in communication systems due to their simple structure and good error correction capabilities.

Recall the Cyclic Codes and its Types

Cyclic codes are a special class of linear block codes that are characterized by the property that the codewords are periodic with a period equal to the length of the codewords. This means that the codewords can be represented as cyclic shifts of a single polynomial.

There are two main types of cyclic codes:

1. Cyclic redundancy check (CRC) codes: CRC codes are a type of cyclic code that are used to detect errors in transmitted messages or signals. They work by adding redundant bits to the transmitted data and using a polynomial division to check the parity of the received data. If the parity check fails, it indicates that an error has occurred in the transmission of the data.

2. BCH codes: BCH codes are a type of cyclic code that are used for error correction. They are characterized by the property that they can correct a certain number of errors for a given code length. BCH codes are widely used in communication systems due to their good error correction capabilities.

Cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the generation of Non-Systematic Cyclic Code

Non-systematic cyclic codes are a type of cyclic code in which the codewords are not the same as the original message or signal. They are generated by multiplying the message or signal by a generator polynomial g(x) to produce the codeword:

c(x) = m(x) * g(x)

where c(x) is the codeword, m(x) is the message or signal, and g(x) is the generator polynomial.

The generator polynomial is chosen such that it has a degree that is equal to or greater than the number of errors that the code is capable of correcting. For example, a generator polynomial of degree 3 can be used to generate a cyclic code that is capable of correcting up to 3 errors.

To decode a received codeword r(x), it is divided by the generator polynomial using polynomial division:

r(x) = q(x) * g(x) + r'(x)

where q(x) is the quotient, g(x) is the generator polynomial, and r'(x) is the remainder. If the remainder r'(x) is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the remainder to identify and correct the errors in the received codeword. If the remainder is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Non-systematic cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the generation of Systematic Cyclic Code

Systematic cyclic codes are a type of cyclic code in which the codewords are the same as the original message or signal, except for the addition of redundant bits. They are generated by adding redundant bits to the message or signal to produce the codeword.

The redundant bits are calculated using a generator polynomial g(x) such that the codeword c(x) satisfies the following equation:

c(x) = m(x) * xk + r(x)

where c(x) is the codeword, m(x) is the message or signal, k is the number of redundant bits, and r(x) is the remainder obtained by dividing m(x) by g(x).

To decode a received codeword r(x), it is divided by the generator polynomial using polynomial division:

r(x) = q(x) * g(x) + r'(x)

where q(x) is the quotient, g(x) is the generator polynomial, and r'(x) is the remainder. If the remainder r'(x) is non-zero, it indicates that errors have occurred in the transmission of the codeword. The decoder can then use the remainder to identify and correct the errors in the received codeword. If the remainder is zero, it indicates that the codeword has been received without errors and the original message or signal can be recovered.

Systematic cyclic codes are widely used in communication systems due to their simple structure and good error correction capabilities. They are particularly useful for detecting errors in transmitted messages or signals, as they can detect a large number of errors with a relatively small overhead.

Describe the Generator and Parity Check Matrix of Cyclic Code

The generator matrix and parity-check matrix are two important concepts in the design and analysis of cyclic codes. They are used to represent the structure of a code and to facilitate the encoding and decoding of messages or signals.

A generator matrix for a cyclic code is a matrix that defines the set of codewords in the code. It is used to encode messages or signals by multiplying the message by the generator matrix to produce the codeword.

The generator matrix for a systematic cyclic code can be written as:

G = [Ik | P]

where Ik is the identity matrix of size k, P is the parity matrix, and k is the number of information bits in the codeword.

A parity-check matrix for a cyclic code is a matrix that is used to check the parity of a received codeword. It is used to decode messages or signals by multiplying the received codeword by the parity-check matrix and checking the resulting syndrome.

The parity-check matrix for a cyclic code can be written as:

H = [PT | I{n-k}]

where PT is the transpose of the parity matrix, I{n-k} is the identity matrix of size n-k, and n is the length of the codewords.

Both the generator matrix and the parity-check matrix are used to represent the structure of a cyclic code and to facilitate the encoding and decoding of messages or signals. They are important tools in the design and analysis of cyclic codes and are widely used in communication systems.

Define the Convolution Codes

Convolutional codes are a type of error correction code that are characterized by the property that the codewords are generated by the convolution of the message or signal with a convolutional encoder. They are widely used in communication systems due to their good error correction capabilities and their ability to handle variable-length messages or signals.

In a convolutional code, the codeword is generated by the convolution of the message or signal with a convolutional encoder, which is a finite state machine that generates the codeword bits based on the current state of the machine and the input message or signal bits. The codeword bits are generated according to a set of rules or transitions defined by the convolutional encoder.

Convolutional codes are characterized by their constraint length K and their rate R, which are defined as follows:

· Constraint length K: The constraint length of a convolutional code is the maximum number of input bits that the encoder takes into account when generating a single output bit.

· Rate R: The rate of a convolutional code is the ratio of the number of output bits to the number of input bits. It is a measure of the efficiency of the code, as it determines the amount of redundancy added to the message or signal.

Convolutional codes are widely used in communication systems due to their good error correction capabilities and their ability to handle variable-length messages or signals. They are particularly useful for correcting errors in long messages or signals, as they can correct a large number of errors with a relatively small overhead.

Describe the Generator of Convolution Codes

The generator of a convolutional code is a matrix that defines the set of codewords in the code. It is used to encode messages or signals by multiplying the message by the generator to produce the codeword.

The generator of a convolutional code can be represented as a polynomial G(D) in the form:

G(D) = g0 + g1D + g2D2 + … + gkDk

where gi are the generator coefficients and D is the delay operator. The generator polynomial is of degree k, which is the constraint length of the code.

The generator polynomial defines the set of codewords in the convolutional code. To encode a message or signal, the message is first represented as a vector m, and the codeword c is obtained by multiplying the message vector by the generator polynomial:

c = mG(D)

where c is the codeword, m is the message vector, and G(D) is the generator polynomial.

The generator of a convolutional code is an important concept in the design and analysis of convolutional codes. It is used to represent the structure of the code and to facilitate the encoding and decoding of messages or signals.

Describe the Graphical Representation of Convolution Codes Encoder

Convolutional codes can be represented graphically using a trellis diagram, which provides a visual representation of the encoding process. The trellis diagram consists of nodes and branches that represent the states and transitions of the convolutional code.

Here’s a graphical representation of a convolutional code encoder using a trellis diagram:

In this example, the trellis diagram represents a convolutional code with a constraint length of 2, meaning it takes into account the current input bit and the previous input bit to determine the encoded output. The nodes represent the possible states of the encoder, while the branches represent the possible transitions between states based on the input bit.

The branches are labeled with two numbers separated by a slash (/). The first number represents the input bit that leads to the transition, and the second number represents the encoded output bit associated with that transition.

Trellis Path for the state diagram: