Question

a. If a binary linear \((n, 2)\) -code corrects one error, show that \(n \geq 5\). [Hint: Hamming bound.] b. Find a (5,2) -code that corrects one error.

Short answer

An (n,2)-code correcting one error requires \(n \geq 5\). Example: code words \(00000\) and \(11111\) for \(n=5\).

Step-by-step solution

Understanding the Hamming Bound

The Hamming bound provides a condition that relates the parameters of a code, including the block length \(n\), the code size \(M\), and the minimum distance \(d\). For a binary code that can correct up to \(t\) errors, the minimum hamming distance \(d\) must satisfy \(d \geq 2t + 1\). If \(t = 1\), then \(d \geq 3\).

Applying the Hamming Bound to (n,2)-Code

A binary linear \((n, 2)\)-code has \(2\) code words. The minimum Hamming distance required to correct one error is at least \(3\) since \(d \geq 3\). The Hamming bound is given by: \[ 2^n \geq 2 \sum_{i=0}^{1} \binom{n}{i} = 2(n+1). \]

Solving the Inequality for n

To satisfy the Hamming bound, we need \(2^n \geq 2(n+1)\). By testing integers, for \(n=5\): \(2^5 = 32\) and \(2(5+1) = 12\). Therefore, \(32 \geq 12\) holds true. Thus, for a code to correct one error, \(n \geq 5\).

Finding a (5,2)-Code That Corrects One Error

A (5,2)-code with a minimum distance of at least 3 can be constructed as follows:Define the code words as: \(00000\) and \(11111\). The minimum distance between these code words is 5, which suffices for correcting one error.

Key concepts

binary linear code

Binary linear codes are a fascinating aspect of coding theory. These codes are constructed from binary vectors, meaning they use only the binary digits 0 and 1. The structure of a binary linear code is based on linear algebra. This is because any linear combination of code words (vectors) from the code is still a code word.For a binary linear code denoted by \(n, M\), \(n\) represents the length of each code word, while \(M\) stands for the number of code words. The primary advantage of these codes is their capacity to detect and correct errors in data transmission. This ability comes from the mathematical structure that governs these codes.Binary linear codes are frequently employed in computer science and telecommunications. They ensure data integrity by making it possible to detect and correct errors without the need for data retransmission. Benign errors may creep during transmission, but thanks to linear codes, these errors can be detected and remedied efficiently.

error correction in coding theory

Error correction is a crucial component of coding theory. It is the process of identifying and rectifying errors in transmitted data to ensure that the received data is as accurate as the original message. This process is vital in scenarios where flawless data transmission is required, such as in communications and data storage.Coding theory provides various methods for error correction. Among these methods, linear codes play an essential role due to their structured nature. The capability to correct errors depends on the minimum Hamming distance between the code words. The larger this distance, the more errors can be detected and corrected.

• For the capability to correct one error, the code must have a minimum Hamming distance of at least 3.
• The Hamming bound is employed to relate parameters like the block length \(n\) and the minimum Hamming distance \(d\).

This bound helps in determining the feasibility of an error correction scheme for a given set of code parameters.

minimum Hamming distance

The minimum Hamming distance is a vital concept in understanding the error-correcting ability of codes. It refers to the smallest number of positions in which any two code words differ. This distance determines how many errors can be detected and corrected.For example, when the minimum Hamming distance is \(d\), a code can detect up to \(d-1\) errors and can correct up to \(\lfloor (d-1)/2 \rfloor\) errors. In simpler terms, a larger Hamming distance implies a better error-correcting capacity.In the context of a binary linear code, for the code to correct at least one error the minimum Hamming distance must be \(d \geq 3\). This condition ensures that even if one error alters a bit in the transmission, the code still allows the original message to be retrieved accurately.Understanding the minimum Hamming distance is crucial for designing efficient error correction mechanisms. It provides a mathematical basis for deciding how many errors can be tolerated and still ensure reliable communication.