REPORTRAJEN KUMAR PATRAROLL NO-14EC62D02 WEEK-1Linear Block Code A linear block code is an error correcting code where the linear combination of a code word is also a code word. It maps the k-bit message sequence into n-bit code word. The k bit message forms 2k distinct message sequence, referred to as k-touples(sequence of k- bits). The n bit block can form as many as 2n distinct sequences, referred to as n-touples. The encoding procedure assigns to each of the 2k message k-touples to one of the 2n n-touples. A block code represents a one to one assignment where the 2k message k-touples are uniquely mapped into a new set of 2k code word n-touples. For linear codes, the mapping is linear. To map the k bit message sequence, we need a generating set of vectors. Any basis set of k linearly independent n-touples V1,V2,. , Vk can be used to generate the required linear block code. That is each of the code word can be described by U= m1V1+ m2V2+m3V3+ + mkVk, where mi =(0 or 1) are message bits and i=1,.,k.So in general, we can define a generator matrix which will be a (kxn) matrix and whose rows will be V1, V2, V3,. ,Vk.

WEEK-2Linear Block Code(Cotd) Linear block code can be systematic which is a mapping from a k-dimensional message bits to an n-dimensional code word in such a way that part of the sequence generator coincides with the k message bits. The remaining (n-k) bits are parity bits. The systematic linear block code will have the generator matrix of the form G=P Ik]where P is the parity array portion of the generator matrix. pij=(0 or 1), and Ik is a (kxk) identity matrix and the systematic code word can be expressed as U= p1,p2,..,pn-k,m1,m2,.,mk where first (n-k) are parity bits and remaining are the message bits.In the receiver end we need a parity check matrix H that will enable us to decode the received vectors. For each (kxn) generator matrix G , there exists an (n-k)xn matrix H such that rows of G are orthogonal to rows of H, that is GHT =0 where HT is the transpose of H. H is a (n-k)x n matrix. For a systematic code, the components of H can be written asH= [In-k PT] WEEK-3

LDPC Code- Low-density parity-check (LDPC) codes are a class of linear block codes. The name comes from the characteristic of their parity-check matrix which contains only a few 1s in comparison to the amount of 0s.

LDPC code can be of two types regular and irregular. In regular LDPC code the number of 1s in the PC matrix is same for every row and every column. In an irregular LDPC code there is no such rule. To represent an irregular LDPC code, we generally take the degree distribution.

LDPC code can be effectively graphically represented by Tanner Graph which provides complete information of the parity check matrix of the graph. Tanner Graphs are bipartite graph where the iterative decoding becomes very easy. Each bit 1 in the parity check matrix is represented by an edge between corresponding variable node (column) and check node (row) of the graph.

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Decoding of LDPC

The decoding of the LDPC code is an iterative process. The decoding can be a Hard Decision Decoding or Soft Decision Decoding. In Hard Decision Decoding, first we take a decision of the received bit (whether it is 0 or 1 ) and then the decoding algorithm happens. Soft Decision Decoding of LDPC codes, which is based on the concept of belief propagation, yields in a better decoding performance and is therefore the preferred method. In soft decision decoding, the LLR(log likelihood ratio) is estimated for each bit and if we take the message passing algorithm, that estimate is passed between check node and bit node.

The Threshold of LDPC code is a very important parameter. The threshold defines an upper bound for the channel noise, up to which the probability of error can be made as small as desired after multiple iterations. Clearly for a good code, threshold should be as high as possible. For irregular code ,we have to find the best possible degree distribution which gives us maximum threshold.