Presentation on theme: "Mahdi Barhoush Mohammad Hanaysheh"— Presentation transcript:
1Mahdi Barhoush Mohammad Hanaysheh Linear Block CodesMahdi BarhoushMohammad Hanaysheh
2Introduction 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) = 2dh (1100,1101) = 1Then 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 guaranteeyou are able to correct a certain number of errors.* dh min = 2*Ct + Dt +1
6Linearity:* 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.
8How to generate a linear code ? Using linear algebra and matrices representation:C = m . GWhereC 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. )
9Ex. LetFind the code for m=SolutionC = m . G =  . G = [ ]
10Ik 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 matrixP is k X (n - k) matrix
11H= [ 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 HSuch that any code C satisfies C . T(H) =[000…0].The above formula implies thatG . T(H) = 0For systematic linear block codeH= [ T(P) I n-k ]
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 integerlet 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
15Hamming code, single error correction: * Suppose that a single error occurred to the transmitted code CIn the i th place, the received vectorr = C + e(i)e(i) : zero row vector with length n except for the i th position.Using the barity check matrix H for decodingr . T(H) = ( C + e(i) ) . T(H) = C . T(H) + e(i) .T(H)= e(i) . T(H) >>> Gives the location of the error