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Standard Array

This document explores the principles of error detection and correction in linear block codes, specifically focusing on valid and invalid codewords, the impact of errors, and the methods for ensuring data integrity during transmission. It discusses the concept of standard arrays, the role of syndromes in decoding, and the importance of minimum distance in code design. Key techniques such as Automatic Repeat Requests (ARQ) and error correction capabilities are examined, alongside the identification of undetected error patterns and the corrective actions taken upon receiving erroneous codewords.

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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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