1. Standard Array

2. 1/3 Repetition Encoder Encoder 2 𝑛 possible combinations of 𝑛 bits
0 0 0 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 𝑘 𝑛
2 𝑛 possible combinations of 𝑛 bits
2 𝑘 valid codewords
2 𝑛 − 2 𝑘 invalid codewords

3. Assume bit 0 is intended to be transmitted
0 1 1
0 0 0
0 0 1
1 1 1
Encoder
0 0 0
0 0 0
0 0 1
1 1 1
0 1 1
Decoder 1
Invalid codeword
Invalid codeword
Valid codeword
1 bit in error
2 bits in error
0 bits in error
3 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
Encoder
0 0 0
1 1 1
0 1 1
Decoder
1 0 0
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
2 𝑘 valid codewords
2 𝑘 -1 undetectable error patterns

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

6. Minimum Distance Error Correction Mode: Error Detection Mode: 0 0 0
dmin is the minimum distance between all the valid codewords
0 0 0
Error Correction Mode:
Error Detection Mode:
𝑑 𝑚𝑖𝑛 −1 2
0 0 1
0 1 0
1 0 0
Error Correction Capability
1 bit in error
1 bit in error
𝑑 𝑚𝑖𝑛 −1
Error Detection Capability
(Correct Correction)
(Detectable)
𝑑 𝑚𝑖𝑛 =3
0 1 1
1 0 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
2 𝑘 valid codewords
0 0 0
1 1 1
0 0 1
1 1 0
0 1 0
1 0 1
1 0 0
0 1 1
2n-k -1 Correctable Error Patterns
Encoder
0 0 0 1
0 1 1
0 1 0
Decoder
0 0 0
1 1 1
Correct Correction
False Correction

8. Linear block Codes (5,2) Linear Block Code 𝑘=2, 𝑛=5𝐺= 𝑘 𝑢 𝑣
0 0
0 1
1 0
1 1
2 2 valid codewords out of 2 5
possible combinations
𝑑 𝑚𝑖𝑛 =3
Error Detection Capability =2
Error Correction Capability =1

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

10. Syndrome Decoding All vectors in a coset have the same symdrome𝒆 𝒖
Encoder 𝒗 𝒓
Decoder 𝒖 𝒗 𝐻=
𝑠=𝑟 𝐻 𝑇
= 𝑣+𝑒 𝐻 𝑇
=𝑣 𝐻 𝑇 +𝑒 𝐻 𝑇
=𝑒 𝐻 𝑇
All vectors in a coset have the same symdrome

11. 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
0 1 0
0 0 1
1 1 0
0 1 1
𝐻 𝑇 =
𝒖=0 1
Encoder 𝒗= 𝒓=
Decoder
𝒖 =0 1
𝑠=1 1 0 𝑒=
𝑣 =

12. 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
0 1 0
0 0 1
1 1 0
0 1 1
𝐻 𝑇 =
1 0 1
1 1 1
For the remaining rows, choose an error pattern that hasn’t appeared before, i.e. with a different syndrome
0 1
1 0
Encoder
Encoder
Decoder
Decoder
1 0
1 0
False Correction
Correct Correction