 Hello everyone welcome to lecture on error correction. At the end of this session students will be able to illustrate error correction in corrupted code words or in error pattern. Now before starting with the actual session let's pause the video and think about what is mean by a syndrome testing. If you remember in the previous slides we already studied that syndrome testing what is in the linear block code. Syndrome testing is nothing but the parity check performed on R. R is nothing but the received vector at the receiver side when you transmit the code word U through a channel in communication system and on that received vector parity check is performed to determine whether the R that is a received vector is a valid member of the code word side which is transmitted from the transmitter through a channel. Now let's start with the error correction. We already seen in the previous sessions that we can detect a single error using a syndrome testing in that we also know that we can perform a syndrome test on both either on the corrupted code word or on the error pattern or on whichever on you perform the syndrome testing that gives you a same syndrome. So from this we can say that it is a clue for us that we can not only detect the error but we can also correct the error which is occurred in the code word which transmitted from the transmitter to the receiver through a channel. So why we can correct the code word easily because there is always a one to one correspondence between the correctable error pattern and the syndrome as we seen already that we get the same syndrome pattern from corrupted code word or on from error pattern by using a syndrome testing. So because of that there is always one to one correspondence between that correctable error pattern and syndrome. So because of this we can correct the error. Now we have a number of code words set. So let's arrange that code word set into 2 res to n n tuples in the array format which is also called as a standard array. For example if you have say 6 tuples so 2 res to 6 that gives you the standard array size. So these are the set code words vectors are arranged in that array format. In that first row is nothing but the all the code words starting with the all zero code word and first column contains the all the correctable error pattern. What are these let's we will see as we go on further. So whatever the figure here we are seeing this is nothing but the standard array format for an n comma k code where n is number of code words and k is nothing but the number of parameters. So here you can see this is nothing but the your first row of your standard array size. So this first row which contains all the code words all the code words starting with this all zero code words. This one is nothing but the having all zero code word and this row column which is nothing but your having a error pattern. So these are the all nothing but the correctable error pattern. So from this we already know that the basic property of linear codes that the all zero code vector must be a member of the code word set. If you see over here these are the first row is nothing but the code word set and all zero this one is nothing but the a member of this code word set. So these are all the code words. Now all the rows in this standard array these are the nothing but the coset. So all rows from the standard array are called as a coset and whatever the first element of that h row or you can say that first column which is having all the elements from each row are called as a coset leader. So each row is nothing but the coset and every element starting element of that row is nothing but the coset leader. Now as we said that the first row is nothing but the code word and first column is nothing but the error pattern correctable error pattern. So from that anyone can say that what about this one first one element which is comes in a first row also and in first column also. So for that this first element is having a dual role or you can say two roles one as I say this one is a valid code word and also it is a act as a correctable error pattern. So like this you one plays two roles first one of the code word these are the code word set from that it is a code word or can also be thought of as a error pattern even. So these are the error patterns which are in the first column so it has two roles means that this one is having R equals to Q means R is a received vector U is a transmitted one mean there is a no error in this one right. Now this array contains all the two rows to N and tuples also it has no repetition or no mistakes in the no missing element in the this one. Now we detected the error now let us decode that one so the decoding algorithm is called for the replacing a corrupted vector with a valid code word from the top of the column containing the corrupted as we seen the standard array in that one if you having a corrupted vector the corresponding top of the column having a corrected code word so you can replace that one with that corrected code word for example suppose that you have a code word which is another having a U which is a transmitted over a noisy channel so because of that noise which results in a error in the received vector which is a R so that R is nothing but your transmitted one U plus having a error that is a plus E E is nothing but the error pattern. So this error pattern is added with your transmitted code word you will get the received vector R which is having a error in that. So if the error pattern is caused by the channel is a coset leader we already seen a coset leader which is nothing but the present in the first column of the standard array so if this error pattern which is added in the transmitted code word is a valid coset leader then the received vector can be decoded correct. How you going to decode that we will going to see in the further slides so let us see what is the syndrome of a coset we already know coset means in nothing but the rows of the standard array right. So if E of j is the coset leader we already seen then U i plus E j is an n tuple in this coset right. So the syndrome for this n tuple can be obtained as s equals to U i plus E j into h t, h of t is nothing but the parity check matrix transpose of the parity check matrix we already seen in the previous session about the parity check matrix. So we have this equation s equals to U i plus E j into h of t. So from that we can rewrite this as U i into h of t plus E j into h of t. So we already know that valid code word in multiply with the parity check matrix transpose of the parity check matrix gives you equals to nothing but the 0. So for this the final equation becomes of the syndrome equals to E of j into h of t right. So using this we can find the syndrome for the error pattern and we have we can find the syndrome for the valid code word set and if both are the same so from that we can easily replace the corrupted vector into the valid code word. So for that the procedure of error correction is that first calculate the syndrome of r using s equal to r into h of t. Here s is the syndrome of the received vector r is nothing but the received vector n h of t is we already seen. Then next is locate the coset leader of error pattern E of j which is nothing but the first element of the each row or you can say the first column of the standard array. Whose syndrome equals to r into h of t right we already calculate the syndrome of the coset in the previous slide. So using that if it is equal to this one we can locate the error pattern. Next step is this error pattern is assumed to be corruption caused by the channel. Next one is the correctable received error vector or a code word is identified as u equals to r into E of j right. So this one is nothing but the u equals to r plus E of j. These are the references. Thank you.