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Magazine Chapter 3: Problem 89
Define a code \(\mathbb{Z}_{2}^{3} \rightarrow \mathbb{Z}_{2}^{6}\) using the standard generator matrix $$G=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right]$$ (a) List all eight code words. (b) Find the associated standard parity check matrix for this code. Is this code (single) error-correcting?
Short Answer
Expert verified
The code has 8 codewords and it is single error-correcting.
Step by step solution
01
Understand the Generator Matrix
The generator matrix \( G \) is a 6x3 matrix that maps 3-bit messages from \( \mathbb{Z}_2^3 \) to 6-bit codewords in \( \mathbb{Z}_2^6 \). Each row corresponds to a new bit added in the codeword. There are 8 possible combinations of 3-bit messages, from \( 000 \) to \( 111 \).
02
Determine the 3-bit Messages
The 3-bit messages are all combinations of binary numbers with 3 bits: \( (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1) \). Each 3-bit message will be multiplied by the generator matrix \( G \) to get the corresponding codeword.
03
Calculate Codewords Using Generator Matrix
Multiply each 3-bit message by the generator matrix \( G \) to obtain the 6-bit codeword. This is computed as follows: Given a 3-bit message vector \( \mathbf{m} = (m_1, m_2, m_3) \), the codeword is \( \mathbf{c} = \mathbf{m} G \).
04
List All Codewords
Perform the matrix multiplication for all 3-bit messages:1. For message \( (0,0,0) \), codeword is \( (0,0,0,0,0,0) \).2. For message \( (0,0,1) \), codeword is \( (0,0,1,0,0,1) \).3. For message \( (0,1,0) \), codeword is \( (0,1,0,0,1,1) \).4. For message \( (0,1,1) \), codeword is \( (0,1,1,0,1,0) \).5. For message \( (1,0,0) \), codeword is \( (1,0,0,1,1,1) \).6. For message \( (1,0,1) \), codeword is \( (1,0,1,1,1,0) \).7. For message \( (1,1,0) \), codeword is \( (1,1,0,1,0,0) \).8. For message \( (1,1,1) \), codeword is \( (1,1,1,1,0,1) \).
05
Understand Parity Check Matrix
The parity check matrix, \( H \), is such that \( G \cdot H^T = 0 \). Since \( G \) is a 6x3 matrix, and has 3 independent columns it's designed to match \( H \)'s 3 dependent columns, such that any error between transmitted code and received code can be detected.
06
Construct Parity Check Matrix
Given that the rows of \( H \) form an orthogonal complement to the rows of \( G \), append an identity matrix of size 3 to the rows of \( G \) as columns to form \( H \):\[ H = \begin{bmatrix}0 & 0 & 0 & 1 & 1 & 1\0 & 0 & 1 & 0 & 0 & 1\0 & 1 & 0 & 0 & 1 & 0\end{bmatrix} \]
07
Check for Error Correction Capability
To determine if this code is a single error-correcting code, verify if for each possible single-bit error, the syndrome \( \mathbf{syndrome} = \mathbf{r} H^T \) provides a unique non-zero value. Since each bit-flip results in a unique non-zero syndrome, this code can correct single errors, making it a single error-correcting code.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Generator Matrix
In the realm of error correction codes, a generator matrix plays a fundamental role. Imagine it as a blueprint that defines how data is encoded for transmission and storage. For the given problem, the generator matrix \( G \) is a 6x3 matrix. This means it's designed to take 3-bit messages from \( \mathbb{Z}_2^3 \) and transform them into 6-bit codewords in \( \mathbb{Z}_2^6 \).
Let's break down its structure and purpose:
Let's break down its structure and purpose:
- Each row of the generator matrix contributes a new bit to the resulting codeword.
- By utilizing matrix multiplication, we can map every possible 3-bit message (there are 8 of them: \( 000 \) to \( 111 \)) to a unique 6-bit codeword. This expands the message space, allowing additional bits that carry redundancy for error detection and correction processes.
- The redundancy is crucial for identifying and correcting errors that may occur during data transmission.
Parity Check Matrix
The parity check matrix, denoted as \( H \), complements the role of the generator matrix by providing a mechanism for identifying errors in codewords. Essentially, it acts as the guardian of data integrity. Here's how it operates:
- The matrix \( H \) is constructed such that the product \( G \cdot H^T = 0 \) holds true, where \( H^T \) is the transpose of \( H \).
- In our given problem, the parity check matrix \( H \) is designed with columns that form an orthogonal complement to the rows of \( G \). This ensures that \( H \) checks for inconsistencies between the transmitted and received code.
- This matrix helps generate a syndrome when codewords are received and checks if there are errors by producing a unique non-zero result when there's a misalignment due to an error.
Matrix Multiplication
Matrix multiplication is the mathematical operation that underpins encoding in error correction codes. It connects our generator matrix \( G \) with an input message vector \( \mathbf{m} \) to produce a codeword \( \mathbf{c} \). Let's dive into the details:
- To encode a 3-bit binary message \( \mathbf{m} \), matrix multiplication involves multiplying \( \mathbf{m} \) with the generator matrix \( G \) to get the 6-bit codeword \( \mathbf{c} = \mathbf{m} G \).
- Consider this process as a series of dot products where each resultant product corresponds to an entry in the resulting codeword vector.
- This technique ensures that all possible messages are consistently and uniquely converted into codewords, allowing for predictable encoding and accurate transmission.
Binary Codewords
Binary codewords are the end product of the encoding process using the generator matrix. They are composed of bits (0s and 1s) that represent the data being transmitted, along with redundant bits that serve error detection and correction purposes.
To put it simply:
To put it simply:
- In this context, each codeword is a 6-bit string resulting from the multiplication of a 3-bit message by the generator matrix \( G \).
- The redundancy added in these 6-bit codewords helps to identify and potentially correct errors during transmission. If any bit undergoes a change due to noise or interference, the additional bits provide enough information to detect and correct this issue.
- The list of generated codewords from the original problem, such as \((0,0,0,0,0,0)\) and \((1,1,1,1,0,1)\), illustrate the mapping from 3-bit messages to comprehensive codewords used in data transmission.
