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Chapter 16: Problem 5

a) What are the dimensions of the generator matrix for the Hamming \((63,57)\) code? What are the dimensions for the associated parity-check matrix \(H ?\) b) What is the rate of this code?

Short Answer

Expert verified
The dimensions of the generator matrix \(G\) for the Hamming \((63,57)\) code are \(57\times63\). The dimensions of the associated parity-check matrix \(H\) are \(6 \times 63\). The rate of this code is \(57/63 = 0.90476\).

Step by step solution

01

Compute dimensions of \(G\)

The generator matrix \(G\) represents the code words of the Hamming code. The dimensions of \(G\) are represented by \(k \times n\). For the Hamming \((63,57)\) code, \(k=57\) and \(n=63\). Therefore, the dimensions of the \(G\) matrix for this code are \(57\times63\).
02

Compute dimensions of \(H\)

The parity-check matrix \(H\) checks the error in the code words. The dimensions of \(H\) are given by \((n-k) \times n\). For the Hamming \((63,57)\) code, \(n=63\) and \(k=57\). Therefore, the dimensions of the \(H\) matrix for this code are \((63-57) \times 63=6 \times 63\).
03

Calculate the rate of the code

The rate of the code \(R\) is given by \(k/n\). For the Hamming \((63,57)\) code, the rate is therefore \(57/63\). This can be simplified to trace the decimal expansion.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Generator Matrix
Understanding the generator matrix is fundamental when working with linear codes like the Hamming code. It's essentially the blueprint for constructing all possible code words of a linear block code. Think of it as a template that, when multiplied with the message bits, generates the encoded bits including the required parity for error detection.

In the case of the Hamming (63,57) code, the generator matrix G has dimensions of 57 by 63. Why these dimensions? Well, the '57' corresponds to the number of message bits, and the '63' represents the total number of bits in the encoded word鈥攎essage bits plus parity bits. By multiplying a 57-bit message vector with the 57 by 63 generator matrix, we obtain a 63-bit code word that follows the structure and properties of the Hamming code.
Parity-Check Matrix
The parity-check matrix H is like a gatekeeper for error detection in code words. This matrix has a direct relationship with the generator matrix G, such that HG = 0 (where '0' represents a zero matrix). For our Hamming (63,57) code, the dimensions of the parity-check matrix H are 6 by 63.

These '6' rows in H are enough to cover the parity bits necessary for the code, while the '63' columns correspond with each bit of the code word. When you multiply the parity-check matrix by a transmitted code word, if no errors are found, the result will be a zero vector. If errors are present, the result helps in identifying and correcting them, leveraging the properties of the Hamming code.
Coding Rate
The coding rate is a measure of efficiency. It tells us how much of our code word is made up of the actual message as opposed to error-checking bits. In a simple formula, the coding rate R is \(R = \frac{k}{n}\), where 'k' is the number of message bits, and 'n' is the total number of bits in the code word.

For the Hamming (63,57) code, we plug in 'k' as 57 and 'n' as 63 to get a rate of \(\frac{57}{63}\), which simplifies to approximately 0.90. This means that 90% of each code word is useful information, and the remaining 10% is dedicated to ensuring error detection and correction capabilities.
Error Detection
The capability of error detection is a key feature of the Hamming code, and it's all about catching and identifying mistakes in transmitted code words. This process relies on both the generator and parity-check matrices. For every code word received, the parity-check matrix is used to calculate a syndrome vector. This syndrome vector is a way of signaling whether an error has occurred.

If the syndrome is a zero vector, then all is well鈥攏o errors detected. But if it's non-zero, the Hamming code taps into its design structure, which is specifically crafted to locate and often correct single-bit errors. The Hamming code鈥檚 method of error detection is very efficient for single-bit errors and some patterns of two-bit errors, offering a robust way to ensure the integrity of data communication.

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Most popular questions from this chapter

For \(n \geq 1\), if \(\sigma, \tau \in S_{n}\), define the distance \(d(\sigma, \tau)\) between \(\sigma\) and \(\tau\) by $$ d(\sigma, \tau)=\max \\{\mid \sigma(i)-\tau(i) \| 1 \leq i \leq n\\} $$ a) Prove that the following properties hold for \(d\). i) \(d(\sigma, \tau) \geq 0\) for all \(\sigma, \tau \in S_{n}\) ii) \(d(\sigma, \tau)=0\) if and only if \(\sigma=\tau\) iii) \(d(\sigma, \tau)=d(\tau, \sigma)\) for all \(\sigma, \tau \in S_{n}\) iv) \(d(\rho, \tau) \leq d(\rho, \sigma)+d(\sigma, \tau)\), for all \(\rho, \sigma, \tau \in S_{n}\) b) Let \(\epsilon\) denote the identity element of \(S_{n}\) (that is, \(\epsilon(i)=i\) for all \(1 \leq i \leq n)\). If \(\pi \in S_{n}\) and \(d(\pi, \epsilon) \leq 1\), what can we say about \(\pi(n) ?\) c) For \(n \geq 1\) let \(a_{n}\) count the number of permutations \(\pi\) in \(S_{n}\), where \(d(\pi, \epsilon) \leq 1\). Find and solve a recurrence relation for \(a_{n}\).

The encoding function \(E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{5}\) is given by the generator matrix $$ G=\left[\begin{array}{lllll} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \end{array}\right] $$ a) Determine all code words. What can we say about the error-detection capability of this code? What about its errorcorrection capability? b) Find the associated parity-check matrix \(H\). c) Use \(H\) to decode each of the following received words. i) 11011 ii) 10101 iii) 11010 iv) 00111 v) 11101 vi) 00110

Let \(G=S_{4}\). (a) For \(\alpha=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{array}\right)\), find the subgroup \(H=\langle\alpha\rangle\). (b) Determine the left cosets of \(H\) in \(G\).

a) Find all generators of the cyclic groups \(\left(\mathbf{Z}_{12},+\right)\), \(\left(\mathbf{Z}_{16},+\right)\), and \(\left(\mathbf{Z}_{24},+\right)\) b) Let \(G=\langle a\rangle\) with \(o(a)=n\). Prove that \(a^{k}, k \in \mathbf{Z}^{+}\), generates \(G\) if and only if \(k\) and \(n\) are relatively prime. c) If \(G\) is a cyclic group of order \(n\), how many distinct generators does it have?

Let \(f: G \rightarrow H\) be a group homomorphism onto \(H\). If \(G\) is a cyclic group, prove that \(H\) is also cyclic.
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