STANDARD ARRAY

1/3 REPETITION ENCODER

Encoder0 0 0 0

1 1 1 1

• valid codewords

• possible combinations of bits

• invalid codewords

0 0 0

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

1 1 1

𝑘 𝑛

Encoder0 0 0 0

• Assume bit 0 is intended to be transmitted

∑ Decoder

0 bits in error

0 0 0

0 0 0

Valid codeword

0

Correct reception

3 bits in error

1 1 1

1 1 1 1

Undetected error

1 bit in error

0 0 1

0 0 1

Invalid codeword2 bits in error

Invalid codeword

0 1 1

0 1 1

• Upon receiving an invalid codeword

• Error Detection (retransmission)

• Error Correction

ERROR DETECTION

Once an invalid codeword is received ask for retransmission

Encoder0 0 0 0

Decoder

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

Automatic Repeat RequestUndetectable Error Pattern:

1 1 1

If the received vector is a valid codeword but not the one intended to be transmitted

valid codewords -1 undetectable error patterns

ERROR CORRECTION

Once an invalid codeword is received attempt to correct it

Encoder0 0 0 0

Decoder

0 0 1

0 1 0

0 1 1

1 0 0

1 0 1

1 1 0

01

Correct CorrectionFalse Correction

MINIMUM DISTANCE

0 0 0

1 1 1

𝑑𝑚𝑖𝑛=3

Error Detection Mode:

1 bit in error

0 0 1 0 1 0 1 0 0

(Detectable)

2 bits in error (Detectable) 0 1 1 1 0 1 1 1 0

3 bits in error (Undetectable)

Error Detection Capability 𝑑𝑚𝑖𝑛−1Error Correction Mode:

1 bit in error (Correct Correction)

2 bits in error (False Correction)

Error Correction Capability

⌊𝑑𝑚𝑖𝑛−12

⌋

• dmin is the minimum distance between all the valid codewords

STANDARD ARRAY

0 0 0 1 1 10 0 1 1 1 0

0 1 0 1 0 1

1 0 0 0 1 1

valid codewords

Encoder0 0 0 0

Decoder0 1 0

0 0 0

0

Correct Correction

0 1 1

1 1 1

1

False Correction

2n-k -1 Correctable Error Patterns

Divide the 2n possible received vectors into 2k regions of valid codewords

LINEAR BLOCK CODES• (5,2) Linear Block Code

𝐺=[1 1 00 1 1

¿1 0¿ 0 1]𝑘𝑛

0 0 0 0 0 0 0

0 1 0 1 1 0 1

1 0 1 1 0 1 0

1 1 1 0 1 1 1

• valid codewords out of

possible combinations

• Error Detection Capability =2

• Error Correction Capability =1

STANDARD ARRAY

0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 11 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 10 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 10 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 10 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 10 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0

valid codewords

Encoder0 1 0 1 1 0 1

Decoder

0 1 1 0 1

0 1

Correct Correction

0 1 1 1 1

Cosets

Coset Leaders

2n-k -1 Correctable Error Patterns

SYNDROME DECODING

Encoder𝒖

Decoder~𝒖𝒗 𝒓

~𝒗∑𝒆

𝐻=[ 1 0 00 1 0

¿1 0¿1 1

0 0 1¿ 0 1 ]

All vectors in a coset have the same symdrome

STANDARD ARRAY

Syndome 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1

1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1

0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1

0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1

1 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1

0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0

Encoder0 1 0 1 1 0 1

Decoder0 10 1 1 1 1

𝐻𝑇=[1 0 00 1 00 0 11 1 00 1 1

]

𝑠=110𝑒=00010~𝑣=01101

STANDARD ARRAY

Syndome 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1

1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1

0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1

0 0 1 0 0 1 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1

1 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 1 0 1 0 1

0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 1 1 0 1 1 0

𝐻𝑇=[1 0 00 1 00 0 11 1 00 1 1

]For the remaining rows, choose an error pattern that hasn’t appeared before, i.e. with a different syndrome

1 0 1 1 0 1 0 00 0 0 1 1

1 1 1 1 0 0 0 10 0 1 1 0

0 1 1 1 0 1 1 0 0 1 1 0 1 0 0

0 1 0 1 1 1 1 1 0 0 1 0 0 0 1

Encoder1 0 1 1 0 1 0

Decoder

1 1 0 1 0

1 0

Correct Correction

1 1 1 0 0Encoder

0 1 0 1 1 0 1Decoder

1 1 0 1 0

1 0

False Correction

1 1 1 0 0∑0 0 1 1 0

∑1 0 0 0 1