1. II. Linear Block Codes

2. Weight Distribution of a Block Code
For an (n.k) block code:
Let Ai be the number of codewords of weight i in C. The numbers A0, A1, …, An are called the
Weight Distribution
Example:
For the (7,4) code shown,
A0 = A7 = 1,
A1 = A2 = A5 = A6= 0
A3 = A4 = 7
Note that:
∑Ai=16=24 (Number of valid
code words)

3. 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
pj(1-p)n-j
In Total, there are nCj error patterns that have j erroneous bits. ONLY Aj of those are NOT DETECTABLE because they represent valid codewords
Probability of not detecting an error pattern (Pu(E)) is:
Example:
For the (7,4) code in the previous slide,
Pu(E)=7p3(1-p)4 + 7p4(1-p)3 + p7

4. Weight Distribution of Coset Leaders
Let αi denote the number of coset leaders of weight i. Then the numbers α0,α1,…, αn are the weight distribution of the coset leaders
Probability of Correct Decoding of an error pattern Ps(E)
PS(E)=∑ αi pi(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-PS(E)

5. Hamming Linear Block Codes

6. Hamming Codes
For any positive integer m≥3, there exists a Hamming code such that:
Code Length: n = 2m-1
No. of information symbols: k = n-m =2m-m-1
No. of parity check symbols: n-k = m
Error correcting capability: t = 1 (i.e., dmin=3)

7. Parity Check Matrix, Generator Matrix and dmin of Hamming Codes
H contains all of nonzero m-tuples as its columns. In the systematic form:
m columns of weight 1
2m-m-1 columns of weight >1
Parity Check Matrix
Generator Matrix
No two columns are identical dmin>2
The sum of any two columns must be a third one dmin=3
Hamming Codes can correct all single errors or detect all double errors

8. Example: (15,11) Hamming Codes
Code Length: n = 24-1 = 15
No. of information symbols: k = 15-4 = 11
No. of parity check symbols: = 4
Error correcting capability: dmin = 3
4 columns of weight 1
11 columns of weight >1

9. Hamming Codes are Perfect Codes
Consider the standard array for a Hamming Code
The code has 2m cosets
The number of (2m-1)-tuples of weight 1 is 2m-1
The 0 vector and the (2m-1)-tuples of weight 1 form all the coset leaders of the standard array
In such configuration for the standard array, the Hamming code corrects ONLY error patterns with single error.
A Perfect Code is a t-error correcting code where its standard array has all error patterns of t or fewer errors and no others as coset leaders

10. The (15,11) Hamming Code is a Perfect Code
Number of cosets = 2n-k= = 16
Number of error patterns of weight 1 = 15
If the coset leaders are chosen to be the 0 vector and all the error patterns of weight 1. The (15,11) code corrects ONLY error patterns with single error

11. Shortened Hamming Codes
It is possible to delete l columns from the parity check matrix of H of a Hamming code.
This results is H’ of dimensions m×(2m-l-1) which is the parity check matrix of a shortened Hamming code
Code Length: n = 2m-l-1
No. of information symbols: k = n-m = 2m-m-l-1
No. of parity check symbols: n-k = m
Error correcting capability: dmin≥3

12. Single Error Correction and Double Error Detection
For a code with minimum Hamming distance dmin:
If the code could correct λ and detect l then dmin≥ λ+l+1
For a shortened Hamming code to achieve single error correction (λ=1) and double error detection (l=2), It is essential that dmin≥4
To achieve dmin=4 it is essential to delete columns properly from H of the original Hamming code
Example
Delete from Q all even weight columns
H’ is of dimension m×2m-1
Why is it that dmin=4?
No two columns are identical  dmin>2
Adding two m-tuples of odd weight results in an even weight m-tuple  dmin>3
Adding three m-tuples of odd weight results in an odd weight m-tuple  dmin=4
2m-1-m odd weight columns

13. Single Error Correction and Double Error Detection (Decoding Scheme)
If the syndrome s is zero, assume no error has occurred.
If s is nonzero and has odd weight, assume single error has occurred. Add the error pattern corresponding to the syndrome.
If s is nonzero and has even weight, an uncorrectable pattern has been detected

14. Example: (8,4) Shortened Hamming Code
Parity Check Matrix for (15,11) Hamming Code
delete all even weight columns

15. Example: (8,4) Shortened Hamming Code
syndrome
s=e.H’T
Notice that
If an error pattern has a single error, the syndrome has odd weight
If an error pattern has a double error, the syndrome has even weight