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Syndrome Decoding of Linear Block Code

Syndrome Decoding of Linear Block Code. Syndrome decoding is a more efficient method of decoding a linear code over a noisy channel ‏ . Syndrome decoding is minimum distance decoding using a reduced lookup table. Prerequisites : Knowledge of Linear Block Codes

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Syndrome Decoding of Linear Block Code

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  1. Syndrome Decoding of Linear Block Code • Syndrome decoding is a more efficient method of decoding a linear code over a noisy channel‏. Syndrome decoding is minimum distance decoding using a reduced lookup table. • Prerequisites : Knowledge of Linear Block Codes • Course Name: Error Correcting Codes Level(UG/PG): PG • Author : Phani Swathi Chitta • Mentor: Prof. Saravanan Vijayakumaran *The contents in this ppt are licensed under Creative Commons Attribution-NonCommercial-ShareAlike 2.5 India license

  2. Learning Objectives After interacting with this Learning Object, the learner will be able to: • Explain the syndrome decoding of a linear block code using the standard array

  3. Definitions of the components/Keywords: 1 • A block code of length and codewords is called a linear (n,k) code if and only if its • codewords form a - dimensional subspace of the vector space of all the - tuples • over the field GF(2). • Any codeword v = uG • where u is the message and G is the generator matrix • The dimension of v is 1 x n, u is 1 x k and G is k x n • The error vector or error pattern e is the difference between the received n-tupler and the transmitted codeword v: • hence the received vector r is the vector sum of the transmitted codeword and the error vector. • r = v + e • When r is received, the decoder computes the following: • s is called the Syndrome of r • The dimension of s is 1 x n-k, r is 1 x n and is n x n-k • Addition of any two codewords results in another codeword and the addition is Modulo – 2 addition 2 3 4 5

  4. Master Layout 1 (6,3) linear block code • Provide a box to enter the received vector that is to be decoded • The vector is sequence of 1s and 0s 2 3 4 Standard Array 5

  5. Step 1: Standard Array Decoding 1 The generator matrix for (6,3) linear code is : G = Generator Matrix 2 3 4 5

  6. Step 2: 1 All the code words of (6,3) are C = { 000 000, 001 011, 010 110, 100 101, 011 101, 101 110, 110 011, 111 000} 2 3 4 5

  7. Step 3: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 2 3 4 5

  8. Step 4: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 2 3 4 5

  9. Step 5: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 2 3 4 5

  10. Step 6: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 2 3 4 5

  11. Step 7: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 2 3 4 5

  12. Step 8: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 2 3 4 5

  13. Step 9: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 2 3 4 5

  14. Step 10: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 2 3 4 5

  15. Step 11: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111001 2 3 4 5

  16. Step 12: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111001 000 010 2 3 4 5

  17. Step 13: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 2 3 4 5

  18. Step 14: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010 100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110 110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 2 3 4 5

  19. Step 15: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010 100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110 110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 2 3 4 5

  20. Step 16: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110 110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 2 3 4 5

  21. Step 17: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010 100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110 110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 100 010 2 3 4 5

  22. Step 18: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010 100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110 110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 100 010 101 001 110 100 000 111 111 111 001 100 010 001 011 010 2 3 4 5

  23. Step 19: 1 Let the received vector r be 000 110 2 3 4 5

  24. Step 20: 1 000 000 001 011 010 110 100 101 011 101 101 110 110 011 111 000 000 001 001 010 010 111 100 100 011 100 101 111 110 010 111 001 000 010 001 001 010 100 100 111 011 111 101 100 110 001 111 010 000 100 001 111 010 010 100 001 011 001 101 010110 111 111 100 001 000 000 011 011 110 101 101 010 101 100 110 111 011 110 000 010 000 011 011 000 110110 101 001 101 111 110 100 011 101 000 100 000 101 011 110 110 000 101 111 101 001 110 010 011 011 000 100 010 101 001 110 100 000 111 111 111 001 100 010 001 011 010 Then the received vector r is decoded to be the 010 110 from the Standard array 2 3 4 5

  25. Step 21: Syndrome decoding 1 2 3 4 5

  26. Step 22: 1 • H = • We know r = v + e and • Therefore since = 0 2 3 4 5

  27. Step 23: 1 S == = 2 3 4 5

  28. Step 24: 1 Coset Leader/ error pattern Syndrome 000 000 000 000 001 001 000 010 010 000 100 100 001 000 011 010 000 110 100 000 101 100 010 111 2 3 4 5

  29. Step 25: 1 Suppose r = 000 110 = = 1 1 0 2 3 4 5

  30. Step 25: 1 Then compute where is the estimated codeword r is the received vector e is the error pattern corresponding to the syndrome S Therefore = 000 110 + 010 000 = 010 110 Hence r is decode as 010 110 2 3 4 5

  31. Electrical Engineering Slide 1 Slide 3 Slide 27 Slide 31 Slide 30 Introduction Definitions Analogy Test your understanding (questionnaire)‏ Lets Sum up (summary)‏ Want to know more… (Further Reading)‏ Interactivity: Try it yourself • Provide a box to enter the received vector G = The user should be able to provide the remaining 9 elements with 1s and 0s 31 Credits

  32. Questionnaire 1 1. Given a (10,5) binary linear block code, how many vectors in each row and column does the Standard array have? Answers: a) , b) , c) , d)‏ None 2. How many correctable error patterns are present in a (n , k) binary linear block code? Answers: a) b) c) d)‏ None 2 3 4 5

  33. Questionnaire 1 3. Given H = Given = 011 110 = 111 001 Does and belong to the same coset? Answers: a) No b) Yes Given = 101 101 and = 101 010 Consider a binary linear block code with minimum distance 3. Then can the vectors be in the same coset? Hint: The minimum distance of linear block code is equal to the minimum weight of its non – zero codewords Answers: a) Yes b) No 2 3 4 5

  34. Links for further reading Reference websites: http://users.ece.gatech.edu/~swm/ECE6606/mod11.pdf http://en.wikipedia.org/wiki/Linear_code Books: Error Control Coding – Shu Lin and Daniel J. Costello, Jr., second editon, Pearson Research papers:

  35. Summary • Syndrome decoding is a more efficient method of decoding a linear code over a noisy channel‏. Syndrome decoding is minimum distance decoding using a reduced lookup table. • A block code of length and codewords is called a linear (n,k) code if and only if its codewords form a - dimensional subspace of the vector space of all the - tuples over the field GF(2). • Any codeword v = uG where u is the message and G is the generator matrix The dimension of v is 1 x n, u is 1 x k and G is k x n • The error vector or error pattern e is the difference between the received n-tuple r and the transmitted codeword v: hence the received vector r is the vector sum of the transmitted codeword and the error vector. r = v + e • When r is received, the decoder computes the following: s is called the Syndrome of r The dimension of s is 1 x n-k, r is 1 x n and is n x n-k

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