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Chapter 16: Problem 3

Find the Hamming weight of the indicated word. $$ 00110111011 $$

Short Answer

Expert verified
The Hamming weight is 6.

Step by step solution

01

Understand Hamming Weight

The Hamming weight of a binary word is the number of '1's in the word. It's also known as the population count or popcount.
02

Count the '1's in the Binary Word

Examine the binary word and count each occurrence of the digit '1'. The given binary word is 00110111011.
03

Apply the Count

In the given binary word, identify the positions with '1': - The digit '1' appears six times. Therefore, the Hamming weight is six.

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Key Concepts

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

Binary Word
A binary word is a string of digits where each digit is a bit that can have a value of either '0' or '1'. It is the fundamental representation of data in computing. Binary words are used to store data, perform calculations, and represent various information in digital systems.

There are several characteristics of binary words that are important to understand:
  • They can vary in length. A binary word might be as short as a few bits or as long as required by a given system.
  • The position of each bit matters; it contributes differently to the value of the binary word. In traditional binary representation, each successive bit from right to left represents increasing powers of two.
  • Binary words are foundational to numerous algorithms and processes in computer science, such as data encoding, encryption, and error detection.
Understanding binary words is crucial for executing operations like measuring their Hamming weight.
Population Count
The population count, also known as popcount, refers to the total number of '1's present within a binary word. It is a term commonly used in programming and computer science.
  • Population count is synonymous with the Hamming weight in binary words.
  • It is essential for various applications. For example, checking error detection in data transmissions often involves calculating the Hamming distance, which requires knowing the Hamming weight.
  • Popcount is also relevant in optimization processes to manage data efficiently, influencing decisions like memory allocation and data compression.
By counting the '1's, we can easily ascertain the binary word's population count and draw conclusions about its data characteristics.
Counting '1's in Binary
Counting the '1's in a binary string is a straightforward process, but it's a crucial step in several important computational tasks. This count gives you the Hamming weight or population count of the binary word.

Here's how to count '1's efficiently:
  • Examine each bit of the binary word from left to right or right to left. Consistency in your method helps to avoid errors.
  • Increment a count each time a '1' is found. This ensures that each '1' is accounted for accurately.
  • Stop once you鈥檝e traversed the entire binary word.
This simple task of counting '1's lays the groundwork for understanding more advanced concepts such as error correction and encoding techniques. Additionally, many programming languages provide built-in functions to calculate population count quickly, aiding in computational efficiency.

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Most popular questions from this chapter

Find the Hamming weight of the indicated word. $$ 01011101011 $$

Show that if \(C\) is an \((n, 3)\) linear binary code that corrects two errors, then \(n \geq 9\).

Find the Hamming distance \(d(u, v)\) between the indicated words. $$ u=000111010101, v=101010101010 $$

Determine the error-detecting capability and the error-correcting capability of the indicated codes, if nearest neighbor decoding is used. $$ \begin{array}{cccc} \text { \(C\)consisting of } & & & \\ 000000 & 001111 & 010111 & 110101 \\ 011000 & 111010 & 100010 & 101101 \end{array} $$

Determine the \((n, k)\) linear binary code with the indicated standard generator matrix. $$ G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end{array}\right] $$
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