memoryless channel.Lety=0,1)be the received value corresponding to the ith variable node.Gallager's hard-decoding algorithm may be deseribed as follows Gallager's hard-decoding [also called Gallager's algorithm B]: Do the following alternatively: l)Update for q.,:For all edges e=(c,s,),l≤i≤n,l≤j≤q,do the following in parallel: If this is the zeroth round,then set=y.Otherwise compute as follows. If more than (-1)of incoming messages along edges in E(e)were equal to the same value be,1)in the previous round.then set=b. Otherwise set q.=y 2)Update forFor all edges e=(c.s).Isisn,Isjsq,do the following in es nodedheete XoR o round along edges in E)eie= 9,(mod2). Note that the message passed contain only extrinsic information;ie.,the value of depends only on the valuewhereruns over all check nodes incident oncother thans (Similarly,for) Decoding is stopped after a preset number of rounds is reached,or before that,provided zH=0.(Each variable node can determine its most likely value based on its neighbors.)If zH0after a preset number of rounds is reached,then we say that the decoding has failed. Note:All message-passing algorithms must respect the which was introduced with turbo codes Rule [Extrinsic information principle:A message sent from a node along an edge e canot depend on any message previously received on edge e. This message passing is to prevent correlation between consecutive decoding iterations. 4.8.3.4 Weighted BF Decoding The simple BF decoding given in Section 4.8.3.2 can be improved by including reliability information of the received symbols in their decoding decisions. Consider the soft-decision received sequence y=()at the output of the 1515 memoryless channel. Let yi∈M={0, 1} be the received value corresponding to the ith variable node. Gallager’s hard-decoding algorithm may be described as follows. „ Gallager’s hard-decoding [also called Gallager’s algorithm B]: Do the following alternatively: 1) Update for i j , q : For all edges ( , ), 1 , 1 i j e cs i n j q = ≤≤ ≤ ≤ , do the following in parallel: If this is the zeroth round, then set ij i , q y = . Otherwise compute i j , q as follows. - If more than δ(|Ev(i)|-1) of incoming messages along edges in Ev(i)\{e} were equal to the same value b∈{0, 1} in the previous round, then set i j , q b = . - Otherwise set ij i , q y = . 2) Update for σ j i, : For all edges ( , ), 1 , 1 i j e cs i n j q = ≤≤ ≤ ≤ , do the following in parallel: The check node sj sends the variable node ci the XOR of the values it received in this round along edges in Ec(j)\{e}; i.e., , ', ' ( )\{ } c ji i j i ji σ q ∈ = ∑ N (mod 2). Note that the message passed contain only extrinsic information; i.e., the value of i j , q depends only on the value σ j i', , where j’ runs over all check nodes incident on ci other than sj. (Similarly, for σ j i, .) Decoding is stopped after a preset number of rounds is reached, or before that, provided T zH 0 = . (Each variable node can determine its most likely value based on its neighbors.) If T zH 0 ≠ after a preset number of rounds is reached, then we say that the decoding has failed. Note: All message-passing algorithms must respect the following rule, which was introduced with turbo codes. Rule [Extrinsic information principle]: A message sent from a node along an edge e cannot depend on any message previously received on edge e. This message passing is to prevent correlation between consecutive decoding iterations. 4.8.3.4 Weighted BF Decoding The simple BF decoding given in Section 4.8.3.2 can be improved by including reliability information of the received symbols in their decoding decisions. Consider the soft-decision received sequence 01 1 ( , ,., ) n y y y = − y at the output of the