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(a) Show that if d = 2t+1 then this code can correct up to t errors. Hint: Use the triangle inequality and show that the distance of the received codeword to other codewords (different form the sent one) is larger than the difference to the originally sent codeword. (b) Is it possible that this code can correct more than t errors when the syndrome decoding is used, explain.

(a) Show that if d = 2t+1 then this code can correct up to t errors. Hint: Use the triangle inequality and show that the distance of the received codeword to other codewords (different form the sent one) is larger than the difference to the originally sent codeword. (b) Is it possible that this code can correct more than t errors when the syndrome decoding is used, explain.

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Problems a) and b).

5. A linear code with parameters [N, K, d] (denoting the length, dimension and minimum distance respectively) has a capability of correcting t errors. (a) Show that if d = 2t + 1 then this code can correct up to t errors. Hint: Use the triangle inequality and show that the distance of the received codeword to other codewords (different form the sent one) is larger than the difference to the originally sent codeword. (b) Is it possible that this code can correct more than t errors when the syndrome decoding is used, explain. (c) The Hamming code [7,4,3] can correct a single error. Can it detect 2 errors when used for error correction, explain ?
Transcribed Image Text:5. A linear code with parameters [N, K, d] (denoting the length, dimension and minimum distance respectively) has a capability of correcting t errors. (a) Show that if d = 2t + 1 then this code can correct up to t errors. Hint: Use the triangle inequality and show that the distance of the received codeword to other codewords (different form the sent one) is larger than the difference to the originally sent codeword. (b) Is it possible that this code can correct more than t errors when the syndrome decoding is used, explain. (c) The Hamming code [7,4,3] can correct a single error. Can it detect 2 errors when used for error correction, explain ?
(d) Specify the generator matrix of the code of length N = equations: 6 given by the following parity check 241 +213 +215 = 0 U₂+U3+ U₁+U6 = 0 Decide what is its minimum distance by considering the parity check matrix built using the above equations.
Transcribed Image Text:(d) Specify the generator matrix of the code of length N = equations: 6 given by the following parity check 241 +213 +215 = 0 U₂+U3+ U₁+U6 = 0 Decide what is its minimum distance by considering the parity check matrix built using the above equations.
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