1. Mahdi Barhoush Mohammad Hanaysheh
Linear Block Codes
Mahdi Barhoush
Mohammad Hanaysheh

2. Introduction to Block Codes
* Block codes introduce controlled amounts of redundancy into a transmitted data stream.
* Block code systems divide uncoded data stream into fixed size blocks (k symbol), then add redundancy to each block (n-k symbols) to form the encoded data stream.
* (k information symbol )+( n-k parity check symbol) >>>
>>>> (n symbol code word)
* Parity or redundancy bits are used for error detection and correction.

4. * Here, we have (n, k) code with rate R = k / n.
* If the symbol is either 0 or 1 >>> Binary block code, symbols are named bits.
* There are 2^n possible code words in a binary block code of length n.
* From these, we choose 2^k code words to be mapped to M = 2^k different message.
* Thus, a block of k information bits is mapped into a code word of length n selected from the set M = 2 ^ k code words.
* Any code has a weight which is the number of nonzero elements that it contains.

5. (measure of difference between any two code words)
* Hamming distance is the number of differences between the corresponding elements in any two code words.
(measure of difference between any two code words)
Ex. dh (1100,1111) = 2
dh (1100,1101) = 1
Then 1100 is closer to 1101
* The smallest hamming distance between any two code words is called the minimum hamming distance dh min.
* The idea with error correction codes is to pick the 2^k code words of the 2^n total possible code words which are far enough apart (in terms of Hamming distance) to guarantee
you are able to correct a certain number of errors.
* dh min = 2*Ct + Dt +1

6. Linearity:
* The block code is called linear block code if the addition of any two code words is also a code word.
* The addition is performed under Galois Field GF(2) in binary block code.
*Linearity implies that the linear block code must contain the all zeros code word.

8. How to generate a linear code ?
Using linear algebra and matrices representation:
C = m . G
Where
C is 1 X n code matrix.
m is 1 X k message matrix.
G is k X n Generation matrix of the code (The rows of the generator matrix are linearly independent and so G has a rank k. )

9. Ex. Let
Find the code for m=[1001]
Solution
C = m . G = [1001] . G = [ ]

10. Ik is the identity matrix P is k X (n - k) matrix
Linear systematic block code:
In symmetric form, the code word C is compromised of a k information segment and a set of n-k symbols that are linear combination of certain information symbols determined by the P matrix.
m = [m1 m2 m3 m4 ] >>> C = [C1 C2 C3 C4 C5 C6 C7]
= [m1 m2 m3 m4 C5 C6 C7]
Then, G = [ Ik P]
Ik is the identity matrix
P is k X (n - k) matrix

11. H= [ T(P) I n-k ] Parity check matrix H: G . T(H) = 0
The (n, k) linear code can be also specified by an (n –k) X n matrix H
Such that any code C satisfies C . T(H) =[000…0].
The above formula implies that
G . T(H) = 0
For systematic linear block code
H= [ T(P) I n-k ]

12. Example.

13. Types of liner block codes:
1. Hamming code.
2. Cyclic Codes.
3. Polynomial Codes.
4. Reed Solomon Codes.
5. Algebraic Geometric Codes.
6. Reed-Muller Codes.
7. Perfect Codes.
8. Repetition Codes.

14. * (n, k) = ( 2^w -1, 2^w -1- w), w is any positive integer
Hamming code:
* (n, k) = ( 2^w -1, 2^w -1- w), w is any positive integer
let w =3 >>>> (7, 4) hamming code
* In Hamming code, the parity check matrix H ,the n column consists of all possible binary vectors except the all zero vectors.
* dh min = 3 >>> can correct one error.
* We have 2^k =16 codes such that C . T(H) = 0

15. Hamming code, single error correction:
* Suppose that a single error occurred to the transmitted code C
In the i th place, the received vector
r = C + e(i)
e(i) : zero row vector with length n except for the i th position.
Using the barity check matrix H for decoding
r . T(H) = ( C + e(i) ) . T(H) = C . T(H) + e(i) .T(H)
= e(i) . T(H) >>> Gives the location of the error