STANDARD ARRAY 13 REPETITION ENCODER 0 1 Encoder

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STANDARD ARRAY

STANDARD ARRAY

1/3 REPETITION ENCODER 0 1 Encoder 000 111 000 001 010 011 100 101

1/3 REPETITION ENCODER 0 1 Encoder 000 111 000 001 010 011 100 101 110 111

• Assume bit 0 is intended to be transmitted 000 1101 0 Encoder

• Assume bit 0 is intended to be transmitted 000 1101 0 Encoder 000 101 0 011 01 Decoder 01 Invalid Valid codeword 2 bits in error 1 bit in error 3 bits in error 0 bits in error codeword Undetected Correct error reception • Upon receiving an invalid codeword • Error Detection (retransmission) • Error Correction

ERROR DETECTION Once an invalid codeword is received ask for retransmission 001 0 Encoder

ERROR DETECTION Once an invalid codeword is received ask for retransmission 001 0 Encoder 010 1 01 11 1 000 100 Decoder 101 110 Repeat Request Undetectable Error. Automatic Pattern: If the received vector is a valid codeword but not the one intended to be transmitted

ERROR CORRECTION Once an invalid codeword is received attempt to correct it 0 Encoder

ERROR CORRECTION Once an invalid codeword is received attempt to correct it 0 Encoder 000 00 10 11 1 00 11 0 00 11 False Correction Decoder 0 1

MINIMUM DISTANCE • dmin is the minimum distance between all the valid codewords Error.

MINIMUM DISTANCE • dmin is the minimum distance between all the valid codewords Error. Correction Detection Mode: Error Mode: Error Correction 1 Error Detection bit error (Detectable) 1 bit inin error (Correction) Capability 2 bits in error (False Correction) (Detectable) 3 bits in error (Undetectable) 000 001 010 100 011 101 110 111

STANDARD ARRAY Divide the 2 n possible received vectors into 2 k regions of

STANDARD ARRAY Divide the 2 n possible received vectors into 2 k regions of valid codewords 2 n-k -1 Correctable Error Patterns 0 Encoder 000 111 0 0 1 1 1 0 0 1 0 1 1 0 0 0 1 1 000 011 0 Decoder 01 01 0 1 Correction False Correction 1 0

LINEAR BLOCK CODES • 00 00000 01 01101 10 11010 11 10111

LINEAR BLOCK CODES • 00 00000 01 01101 10 11010 11 10111

STANDARD ARRAY 2 n-k -1 Correctable Cosets Error Patterns 01 00000 011010 10111 1

STANDARD ARRAY 2 n-k -1 Correctable Cosets Error Patterns 01 00000 011010 10111 1 0 0 11101 01010 00111 0 0 0 00101 10010 11111 0 0 01001 11110 10011 0 0 0 1 0 01111 11000 10101 0 0 1 01100 110110 Coset 01101 Leaders Encoder 01111 Decoder 01101 Correction 01

SYNDROME DECODING Encoder Decoder All vectors in a coset have the same symdrome

SYNDROME DECODING Encoder Decoder All vectors in a coset have the same symdrome

STANDARD ARRAY Syndome 00000 011010 10111 100 1 0 0 11101 01010 00111 010

STANDARD ARRAY Syndome 00000 011010 10111 100 1 0 0 11101 01010 00111 010 0 1 0 00101 10010 11111 001 0 0 01001 11110 10011 110 0 1 0 01111 11000 10101 011 0 0 1 01100 110110 Encoder Decoder

STANDARD ARRAY Syndome 00000 011010 10111 100 1 0 0 11101 01010 00111 010

STANDARD ARRAY Syndome 00000 011010 10111 100 1 0 0 11101 01010 00111 010 0 1 0 00101 10010 11111 001 0 0 01001 11110 10011 110 0 1 0 01111 11000 10101 011 0 0 1 01100 110110 101 0 00 10 1 01110 11001 10100 111 10 00 01 01 10 01011 11100 10001 10 1 01 00 1 0 For the remaining rows, choose an error pattern that hasn’t appeared before, i. e. with a different syndrome 11 0 11100 0 11 1 00 11 0 0 10 Encoder Decoder 11010 Correction False Correction

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