TAILIEUCHUNG - Chapter 4 - Hamming Codes

In the late 1940’s Claude Shannon was developing information theory and coding as a mathematical model for communication. At the same time, Richard Hamming, a colleague of Shannon’s at Bell Laboratories, found a need for error correction in his work on computers. Parity checking was already being used to detect errors in the calculations of the relay-based computers of the day, and Hamming realized that a more sophisticated pattern of parity checking allowed the correction of single errors along with the detection of double errors. The codes that Hamming devised, the single-error-correcting binary Hamming codes and their single-error-correcting, double-error-detecting extended. | Chapter 4 Hamming Codes In the late 1940 s Claude Shannon was developing information theory and coding as a mathematical model for communication. At the same time Richard Hamming a colleague of Shannon s at Bell Laboratories found a need for error correction in his work on computers. Parity checking was already being used to detect errors in the calculations of the relay-based computers of the day and Hamming realized that a more sophisticated pattern of parity checking allowed the correction of single errors along with the detection of double errors. The codes that Hamming devised the single-error-correcting binary Hamming codes and their single-error-correcting double-error-detecting extended versions marked the beginning of coding theory. These codes remain important to this day for theoretical and practical reasons as well as historical. Basics Denote by L3 the check matrix that we have been using to describe the 7 4 Hamming code 0 0 0 1 1 1 1 L3 0 1 1 0 0 1 1 1 0 1 0 1 0 1 It has among its columns each nonzero triple from F2 exactly once. From this and Lemma we were able to prove that the 7 4 Hamming code has minimum distance 3. This suggests a general method for building binary Hamming codes. For any r construct a binary r X 2r 1 matrix H such that each nonzero binary r-tuple occurs exactly once as a column of H. Any code with such a check matrix H is a binary Hamming code of redundancy r denoted Hamr 2 . Thus the 7 4 code is a Hamming code Ham3 2 . Each binary Hamming code has minimum weight and distance 3 since as before there are no columns 0 and no pair of identical columns. That is no pair of columns is linearly dependent while any two columns sum to a third column giving a triple of linearly dependent columns. Lemma again applies. binary Hamming code 49 lexicographic check matrix extended Hamming code 50 CHAPTER 4. HAMMING CODES As defined any code that is equivalent to a binary Hamming code is itself a Hamming code since any .

• Tiếng Anh thương mại
• Hamming Codes
• the message
• data word
• hamming code
• Hamming realized
• calculations of the relay based
• Channel coding
• Cyclic codes
• Block diagram
• Error control techniques
• Lecture Wireless networks
• Wireless networks
• Bài giảng Mạng không dây
• Hamming code process
• Block codes
• Multiple error correcting codes