Basics of Hamming Distance
Duration: 8 min
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This educational video provides a comprehensive introduction to Hamming Distance and its application in coding theory. The instructor begins by defining Hamming Distance as the number of differences between corresponding bits of two words of the same size, denoted as d(x, y). He demonstrates a calculation method using the XOR operation. The lecture then progresses to Minimum Hamming Distance, defined as the smallest distance between any pair of codewords in a set. The instructor calculates this for a specific set of four 5-bit codewords. Finally, the video introduces the three essential parameters of a coding scheme: codeword size (n), dataword size (k), and minimum Hamming distance (d_min). The instructor explains how these parameters are used to describe a coding scheme, using the notation C(n, k), and provides examples based on the calculations performed earlier in the lecture.
Chapters
0:00 – 2:00 00:00-02:00
The lecture begins with the definition of Hamming Distance. The on-screen text states: "The Hamming distance between two words (of the same size) is the number of differences between the corresponding bits." The notation d(x, y) is introduced to represent this distance. The instructor explains a practical method: "The Hamming distance can easily be found if we apply the XOR operation on the two words and count the number of 1s in the result." He demonstrates this with the example d(000, 011) = 2 because 000 XOR 011 = 011 (two 1s). He then presents a problem to solve: "The Hamming distance d(10101, 11110) is _______?" He proceeds to solve this on the whiteboard, writing out the bits and counting the differences, concluding the answer is 3. He explicitly writes d(10101, 11110) = 3 on the board.
2:00 – 5:00 02:00-05:00
The concept of Minimum Hamming Distance is introduced. The definition is given as "the minimum Hamming distance is the smallest Hamming distance between all possible pairs." The instructor lists a set of four codewords: C1 = 00000, C2 = 01011, C3 = 10101, C4 = 11110. He systematically calculates the Hamming distance for every pair to find the minimum. He writes down the calculations: d(C1, C2) = 3, d(C1, C3) = 3, d(C1, C4) = 4, d(C2, C3) = 4, d(C2, C4) = 3, and d(C3, C4) = 3. By comparing these values, he determines that the smallest distance is 3, so d_min = 3. He underlines the value 3 to emphasize it is the minimum.
5:00 – 7:45 05:00-07:45
The final section covers the "Three Parameters" of a coding scheme. The instructor explains that any coding scheme needs at least three parameters: the codeword size n, the dataword size k, and the minimum Hamming distance d_min. He states that a coding scheme C is written as C(n, k) with a separate expression for d_min. He refers back to previous examples to illustrate this. He identifies the first coding scheme (mapping 2-bit datawords to 3-bit codewords) as C(3, 2) with d_min = 2. He identifies the second coding scheme (mapping 2-bit datawords to 5-bit codewords) as C(5, 2) with d_min = 3. He points to tables on the screen showing the specific mappings of datawords to codewords for these schemes, reinforcing the relationship between the parameters.
The video effectively builds a logical progression from basic distance measurement to coding scheme characterization. It starts with the fundamental concept of Hamming Distance, showing how to calculate it between two specific words using bitwise operations. This foundation is then extended to a set of words to define Minimum Hamming Distance, which is crucial for error detection and correction capabilities. The final section synthesizes these concepts by introducing the standard notation C(n, k) and the parameter d_min, linking the abstract definitions to concrete examples of dataword and codeword mappings. This structure helps students understand not just how to calculate distances, but how these distances define the properties of a coding scheme.