1. II. Linear Block Codes

2. © Tallal Elshabrawy 2 Last Lecture H Matrix and Calculation of d min Error Detection Capability Error Correction Capability Error Correction & Error Detection Operation

3. © Tallal Elshabrawy 3 Decoding: A rule to partition 2 n possible received vectors into 2 k disjoint subsets D 1, D 2, …,D 2 k such that the codeword v i is contained in the subset D i, 1≤i ≤2 k Let C be an (n,k) linear block code with codewords v 1, v 2, …,v 2 k Channel v e r Decoder Which v i was sent?? 2 k codewords 2 n n-tuples Decoding Rule: Correct Decoding: Given v i is transmitted, error pattern e does not push r outside D i Decoding and Error Correction

4. © Tallal Elshabrawy 4 Design of a Decoder How to divide 2 n vectors into 2 k disjoint subsets such that the error correcting capability is optimal? In other words, which n-tuples should be designed such that they belong to a given subset D i ?

5. © Tallal Elshabrawy 5 Standard Array Partitioning 2 n possible received vectors into 2 k disjoint subsets with each subset containing only one codeword D1D1 D2D2 DiDi D2kD2k

6. © Tallal Elshabrawy 6 Standard Array Partitioning 2 n possible received vectors into 2 k disjoint subsets with each subset containing only one codeword e 2 ≠v 1,v 2,…,v 2 k D1D1 D2D2 DiDi D2kD2k

7. © Tallal Elshabrawy 7 Standard Array Partitioning 2 n possible received vectors into 2 k disjoint subsets with each subset containing only one codeword e 3 ≠v 1,v 2,…,v 2 k, e 2,…,e 2 +v 2 k D1D1 D2D2 DiDi D2kD2k

8. © Tallal Elshabrawy 8 Standard Array Partitioning 2 n possible received vectors into 2 k disjoint subsets with each subset containing only one codeword e 2 n-k ≠Any previous element in the array D1D1 D2D2 DiDi D2kD2k

9. © Tallal Elshabrawy 9 Theorem 1.No two n-tuples in the same row of a standard array are identical. 2.Every n-tuple appears in one and only one row. PROOF: 1.Given a linear block code is considered. By definition of the standard array construction, e j is not a codeword. Therefore, e j + v j generate distinct patterns 2.Select row l<m, assume e l +v i = e m +v j. Therefore, e m = e l +v i +v j e m = e l +v s (because v i and v j are codewords) This means that e m could be found in row l This is a contradiction to the construction rule of the standard array

10. © Tallal Elshabrawy 10 Standard Array Partitioning 2 n possible received vectors into 2 k disjoint subsets with each subset containing only one codeword e 2 n-k ≠Any previous element in the array D1D1 D2D2 DiDi D2kD2k

11. © Tallal Elshabrawy 11 Coset of a Code, Coset Leader Given the standard array. The 2 n-k rows are called the cosets of the code. The first n-tuple e j of each coset is called the coset leader

12. © Tallal Elshabrawy 12 Error Correction A Linear block code can correct 2 n-k -1 error patterns corresponding to 2 n-k -1 coset leaders Note that one of the coset leaders is the zero vector

13. © Tallal Elshabrawy 13 Error Correction Methodology The error patterns that are most likely to occur should be used as the coset leaders Example: A (7,4) code can correct 2 (7-4) -1=7 error patterns To correct a single error. The coset leaders should be selected as: e 1 = v 1 = (0 0 0 0 0 0 0) e 2 =(0 0 0 0 0 0 1) e 3 =(0 0 0 0 0 1 0) e 4 =(0 0 0 0 1 0 0) e 5 =(0 0 0 1 0 0 0) e 6 =(0 0 1 0 0 0 0) e 7 =(0 1 0 0 0 0 0) e 8 =(1 0 0 0 0 0 0)

14. © Tallal Elshabrawy 14 Theorem 1.All 2 k n-tuples of a coset have the same syndrome. 2.The syndrome for different cosets are different PROOF: 1.(e l +v i )H T =e l H T +v i H T =e l H T (independent of v i ) 2.Let e j and e i be the coset leaders. If they have the same syndrome e j H T +e i H T =0. Therefore, (e j +e i ) is a codeword This contradicts the definition of a linear block code because e j and e i are not codewords Advantages: We can use a syndrome decoding table which is much simpler to use than a standard array

15. © Tallal Elshabrawy 15 Syndrome Decoding 1.Compute syndrome of r (i.e., rH T ) 2.Locate the coset leader e l whose syndrome is equal to rH T 3.v * = r+e l where v * is the codeword output by the decoder

16. © Tallal Elshabrawy 16 Weight Distribution of a Block Code For an (n.k) block code: Let A i be the number of codewords of weight i in C. The numbers A 0, A 1, …, A n are called the Weight Distribution Example: For the (7,4) code shown, A 0 = A 7 = 1, A 1 = A 2 = A 5 = A 6 = 0 A 3 = A 4 = 7 Note that: ∑A i =16=2 4 (Number of valid code words)

17. © Tallal Elshabrawy 17 Weight Distribution and Probability of Detecting an Error Pattern For an (n,k) linear code: Given that the bit error probability of the physical channel is p. The probability that an error pattern of weight j occurs is p j (1-p) n-j In Total, there are n C j error patterns that have j erroneous bits. ONLY A j of those are NOT DETECTABLE because they represent valid codewords Probability of not detecting an error pattern (P u (E)) is: Example: For the (7,4) code in the previous slide, P u (E)=7p 3 (1-p) 4 + 7p 4 (1-p) 3 + p 7

18. © Tallal Elshabrawy 18 Weight Distribution of Coset Leaders Let α i denote the number of coset leaders of weight i. Then the numbers α 0,α 1, …, α n are the weight distribution of the coset leaders Probability of Correct Decoding of an error pattern P s (E) P S (E)=∑ α i p i (1-p) (n-i) where is the bit error probability of the channel Probability of False Decoding of an error pattern P(E) P (E)=1-P S (E)