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# Linear block code

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### Linear block code

1. 1. LINEAR BLOCK CODINGPresented by:Manish Srivastava
2. 2. LINEAR BLOCK CODEIn a (n,k) linear block code:1st portion of k bits is always identical to the message sequence to be transmitted.2nd portion of (n-k ) bits are computed from message bits according to the encoding rule and is called parity bits.
3. 3. SYNDROME DECODING The generator matrix G is used in the encoding operation at the transmitter The parity- check matrix H is used in the decoding operation at the receiver Let , y denote 1-by-n received vector that results from sending the code x over a noisy channel y=x +e
4. 4.  For i=1,2,….., n ei= 1,if an error has occurred in the ith location 0 ,otherwiseo s=yHt
5. 5. PROPERTIESProperty 1: The syndrome depends only on the error pattern and not on the transmitted code word. S=(x+e)Ht =xHt+ eHt =eHt
6. 6. PROPERTY 2: All error pattern that differs at most by a code word have the same syndrome. For k message bits ,there are 2k distinct codes denoted as xi ,i=0,1, ………. 2k -1we define 2k distinct vectors as e =e+ xi i=0,1,…….. 2k-1
7. 7. =e +=e
8. 8. PROPERTY 3: The syndrome s is the sum of those columns of matrix H corresponding to the error locations H=[ , ………., ] therefore, s=
9. 9. PROPERTY 4: With syndrome decoding ,an (n,k) linear block code can correct up to t errors per code word ,provided that n and k satisfy the hamming bound ≥ ( ) where ( ) is a binomial coefficient ,namely ( )= n!/(n-i)!i!
10. 10. MINIMUM DISTANCE CONSIDERATIONS: Consider a pair of code vectors x and y that have the same number of elements Hamming distance d(x,y): It is defined as the number of locations in which their respective elements differ . Hamming weight w(x) : It is defined as the number of elements in the code vector.
11. 11.  Minimum distance dmin: It is defined as the smallest hamming distance between any pair of code vectors in the code or smallest hamming weight of the non zero code vectors in the code .
12. 12.  An (n,k) linear block code has the power to correct all error patterns of weight t or less if ,and only if d( ) ≤2t+1 An (n,k) linear block code of minimum distance dmin can correct upto 1 error if and only if t≤ [1/2 (dmin – 1)].
13. 13. Advantages DisadvantagesEasiest to detect and  Transmission correct errors. bandwidth is more.Extra parity bit does not  Extra bit reduces the convey any information bit rate of transmitter but detects and and also its power. corrects errors.
14. 14. APPLICATIONS Used for error control coding. Storage-magnetic and optical data storage in hard disks and magnetic tapes and single error correcting and double error correcting code(SEC- DEC) used to improve semiconductor memories. Communication-satellite and deep space communications.
15. 15. THANK YOU!!