Basics of Hamming Distance

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This lecture introduces Hamming distance as a fundamental metric in coding theory, defining it as the number of differences between corresponding bits in two words of equal size. The instructor demonstrates calculation methods using XOR operations and bitwise comparison, then extends the concept to Minimum Hamming Distance within a set of codewords. Finally, the lesson establishes three critical parameters for coding schemes: codeword size n, dataword size k, and minimum Hamming distance dmin.

Chapters

1. 0:00 – 2:00 00:00-02:00

   The video opens with a definition of Hamming distance, denoted as d(x, y), representing the count of differing bits between two words. The instructor explains that applying an XOR operation on two binary words and counting the resulting 1s provides a quick calculation method. On-screen text explicitly states: 'The Hamming distance between two words (of the same size) is the number of differences between the corresponding bits.' A worked example demonstrates d(000, 011) equals 2 because the XOR result is 011, containing two 1s. The segment concludes by introducing a practice problem: 'The Hamming distance d(10101, 11110) is ________ ?' requiring students to apply the XOR method.

2. 2:00 – 5:00 02:00-05:00

   The lecture transitions to Minimum Hamming Distance, defined as the smallest distance between any pair of codewords in a set. The instructor presents four binary codewords labeled C1 through C4 and systematically calculates pairwise distances to identify the minimum. Visible calculations include d(00000, 01011) = 3 and d(01011, 11110) = 3. The instructor writes these values on a whiteboard next to the slide content, showing d(01011, 10101) = 4 and d(00000, 11110) = 4. After evaluating all combinations, the instructor concludes that d_min equals 3, emphasizing that this value determines error detection and correction capabilities.

3. 5:00 – 7:46 05:00-07:46

   The final section establishes three essential parameters for any coding scheme: codeword size n, dataword size k, and minimum Hamming distance dmin. The notation C(n, k) is introduced to describe these schemes. Two specific examples are displayed in tables mapping datawords to codewords: C(3, 2) with dmin=2 and C(5, 2) with dmin=3. The instructor points to specific rows in the dataword table and circles parameters n and k in formulas, highlighting how input datawords transform into output codewords. The slide text explicitly lists 'Three Parameters' and shows the relationship between input size, output size, and error resilience.

The lecture progresses logically from basic distance calculation to system-level parameters. First, Hamming distance is defined as a bit-counting metric solvable via XOR operations. Second, Minimum Hamming Distance is derived by comparing all codeword pairs in a set to find the smallest value. Third, these concepts are unified under three coding parameters: n (codeword size), k (dataword size), and dmin. The examples C(3, 2) with dmin=2 and C(5, 2) with dmin=3 illustrate how different configurations affect error handling. The visual evidence shows consistent use of binary strings, XOR logic, and tabular mappings to reinforce these relationships.