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Standard Array

Standard Array. 1/3 Repetition Encoder. 0. 0 0 0. Encoder. 1. 1 1 1. 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1. possible combinations of bits. valid codewords. invalid codewords. Assume bit 0 is intended to be transmitted. 0 1 1. 0 0 0. 1 1 1. 0 0 1. 1 1 1.

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Standard Array

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  1. Standard Array

  2. 1/3 Repetition Encoder 0 0 0 0 Encoder 1 1 1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 • possible combinations of bits • valid codewords • invalid codewords

  3. Assume bit 0 is intended to be transmitted 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 Encoder 0 0 0 Decoder 0 Invalid codeword Invalid codeword Valid codeword 2 bits in error 1 bit in error 3 bits in error 0 bits in error Undetected error Correct reception • Upon receiving an invalid codeword • Error Detection (retransmission) • Error Correction

  4. Error Detection Once an invalid codeword is received ask for retransmission 0 0 1 0 1 0 1 1 1 0 0 0 0 Encoder Decoder 0 1 1 1 00 1 0 1 1 1 0 Automatic Repeat Request Undetectable Error Pattern: If the received vector is a valid codeword but not the one intended to be transmitted valid codewords -1 undetectable error patterns

  5. Error Correction Once an invalid codeword is received attempt to correct it 0 1 1 0 0 1 0 0 1 0 0 0 Encoder Decoder 1 0 1 0 1 0 1 00 1 1 0 False Correction Correct Correction

  6. Minimum Distance • dmin is the minimum distance between all the valid codewords 0 0 0 Error Detection Mode: Error Correction Mode: 0 1 0 0 0 1 10 0 Error Correction Capability 1 bit in error Error Detection Capability (Detectable) 1 bit in error (Correct Correction) 1 0 1 0 1 1 1 1 0 2 bits in error 2 bits in error (False Correction) (Detectable) 3 bits in error (Undetectable) 1 1 1

  7. Standard array Divide the 2n possible received vectors into 2k regions of valid codewords valid codewords 2n-k -1 Correctable Error Patterns 1 0 0 0 0 Encoder Decoder 0 0 1 1 0 1 0 0 0 0 1 1 1 Correct Correction False Correction

  8. Linear block Codes • (5,2) Linear Block Code • valid codewords out of possible combinations • Error Detection Capability =2 • Error Correction Capability =1

  9. Standard Array valid codewords 2n-k -1 Correctable Error Patterns Cosets Coset Leaders 0 1 1 1 1 0 1 0 1 1 0 1 Encoder Decoder 0 1 0 1 1 0 1 Correct Correction

  10. Syndrome Decoding Encoder Decoder All vectors in a coset have the same symdrome

  11. Standard Array 0 1 1 1 1 0 1 0 1 1 0 1 Encoder Decoder 0 1

  12. Standard Array 1 0 1 1 0 1 0 0 0 0 0 11 0 1 11 0 1 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 11 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 For the remaining rows, choose an error pattern that hasn’t appeared before, i.e. with a different syndrome 1 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 01 Encoder Decoder 1 0 Encoder Decoder 1 0 1 1 0 10 1 1 0 10 Correct Correction False Correction

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