LINEAR BLOCK CODINGPresented by:Manish Srivastava
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.
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
For i=1,2,….., n ei= 1,if an error has occurred in the ith location 0 ,otherwiseo s=yHt
PROPERTIESProperty 1: The syndrome depends only on the error pattern and not on the transmitted code word. S=(x+e)Ht =xHt+ eHt =eHt
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
PROPERTY 3: The syndrome s is the sum of those columns of matrix H corresponding to the error locations H=[ , ………., ] therefore, s=
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!
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.
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 .
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)].
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.
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.