 Welcome to lecture on systematic linear blockcodes. At the end of this session, students will be able to describe and construct generator matrix. Now before starting with the actual session, let's pause the video and recall what is mean by linear blockcode. It means that if you have two codewords ui and uj in an n, k blockcode and a1 and a2 are any elements selected, then the code is said to be linear if a1 ui plus a2 uj is also a codeword. If you remember the linear property from the mathematics a1 ax plus by equals to ax by, it is similar to that one. So means that if one codeword is multiplied with the element and added with the other codeword multiplied with the other element, whatever the codeword you are having third one, it is also a codeword. Now let's see what is mean by generator matrix. We know that u equals to m into g, where u is the codeword matrix, m is the information block or you can say data words and g is nothing but the generator matrix. So this can be represented as u equals to u1, u2 up to un. As in the previous slides we seen that u is nothing but the codeword block which is having n distinct codewords, m is nothing but the message block which is having k distinct message words. So m1, m2 up to mk and g is nothing but the generator matrix in this one which is having a row from 1 to k. So if you form the multiplication of this one, m1, m2, mk is multiplied with the g1, g2 up to gk. So you are getting the equation is what? m1 into g1 plus m2 into g2 up to mk into gk. So from that you can say that generally the rows of generator matrix are linearly independent and also they form a base for the n, k loads. Now let's see what is mean by systematic linear block codes. So block code n, k is said to be systematic when a mapping of from k dimensional message vector to an n dimensional codeword is done and that mapping is done in such a way that a part of sequence generated coincides with the k message digits. What is mean this we can see we going to see in the later. So means that whatever the k message digits are same as in the codeword. So remaining n minus k digits are nothing but a parity bits or you can say that redundant bits. This one is you can see u is nothing but your codeword block in that you are having n distinct codewords u1, u2 up to un and that having a combination of message bits and parity bits. So from that from 1 to k these are the message bits and from p1 to n minus k these are the parity bits. So generally for systematic linear block codes a generator matrix having a different form so g equals to pi. So from that p is nothing but your parity bits array and i is nothing but your identity matrix. So due to this the complexity at the encoding is reduced and because of use of identity matrix this one is fixed either you are encoding or decoding identity matrix is fixed so there is no need to store identity matrix we know that what is mean by identity matrix which is having diagonal elements only 1 and non diagonal elements are 0. So this one is a standard form though there is no need to store the identity matrix values. So g equals to pi which is having a size of k by n. So from that from 1 first row up to k row and from first column up to n minus k column these are nothing but the parity bits and from n minus k plus 1 to n these are nothing but the identity matrix. So you can say that so parity array is a size of k by n minus k whereas the identity matrix is having size of k by k we know that the identity matrix has nothing but which is having same row and same column numbers. Now for message vector m equals to m1 m2 up to mk and codeword which is having u1 u2 un let us get the generator matrix. So we know that u equals to nothing but the m into g. So u equals to nothing but we already said codeword equals to u1 u2 up to n because we know that codewords having n distincts codewords and m is nothing but m1 m2 up to k. Ok distinct message words g we already seen in the previous slide which is having nothing but the combination of parity matrix parity matrix and identity matrix. So which is having this p1 1 p2 1 and up to pk1 and column having p1 n minus k up to row pk n minus k. So this is nothing but the parity array and these are this is nothing but the identity matrix which is having diagonal elements 1 and non-diagonal elements are 0. So form that if you perform the multiplication we get the general formula for u that is codeword. So in general formula is what ui equals to m1 p1 i plus m2 p2 i plus up to mk pk i. So where i is for range of from 1 to n minus k because we know that from generator matrix we having up to n minus k are the parity array and from n minus k plus 1 to n we having a identity matrix. So form because of that whatever the codeword you are getting over here these are the having this formula and after n minus k plus 1 to up to n whatever the codeword you are getting is nothing but the same as your message bits. So we can say that for i equals to n minus k plus 1 up to n code bits will be the same as the message bits because of this is a systematic linear block code. Why we are saying systematic linear block code? Because of generator matrix whatever we are using in this one is having special form which is forms by using a parity array and identity matrix. So because of that identity matrix whatever the code bits we are getting for n minus k plus 1 up to n are same as the message bits. So from that we can say that in other terms a equals to u 1 plus u 2 plus up to u n minus k and after u n minus k plus 1 up to u n. So this is a total your codeword and from that because of this one is a systematic linear block code up to n minus k these are the parity bits. And after this from n minus k plus 1 to n these are the message bits because from this in the generator matrix we are having identity matrix. So same can be expressed as p 1 p 2 p i up to p n minus k and after this we are having a message bits so m n minus k plus 1 up to m n. So these are the parity bits and these are the message bits. So from that we can easily get the formula for parity bits also the equation is same as we derived for the code bits in the previous two slides. So p 1 equals to m 1 p 1 1 plus m 2 p 2 1 plus up to m k p k 1 similarly up to p n minus k because we know that the parity array is having a size of from p 1 to p n minus k. So p n minus k equals to m 1 p 1 n minus k up to m k p k n minus k. So these parity bits p 1 to p n minus k are exactly same as the first n minus k elements of the codeword of vector u. Now let us see the example consider a 6 comma 3 block code generated by generator matrix G G equals to given over here 1 0 1 1 1 0 0 1 1 1 0 0 0 1 0 and 0 0 1. So this one is a 6 comma 3 6 columns and 3 rows. So from that we say that this one is format is n comma k so n is nothing but the 6 and k is nothing but the 3. So before finding the codewords we should know that how many data words or how many message bits is offered. So that can be obtained by 2 raise to k so 2 raise to k k over here is a 3 so 2 raise to 3 equals to 8. So there are 8 data words possible. So these are nothing but 8 data words 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 up to 1 1 1. So now we are going to find the codewords for each data word. So for that the same can be done in the previous session we have seen in the introduction to linear block code but in that one generator matrix is not having any special form. In this one this one is a systematic linear block code so generator matrix having a special form means that it is having a parity array and which is combined with the identity matrix. So because of that identity matrix whatever the codewords you are getting that codewords having a same code bits as your message bits because of the identity matrix. So we already derived the formula for that. So now let us see what is the actual equation. So u equals to m1 m2 m3 is multiplied with the generator matrix we already know the formula for the u u equals to m into g m is nothing but your message words and g is nothing but the generator matrix. So if you take a first bit 0 0 0 is multiplied with the this generator matrix from that this first 3 columns are nothing but the parity array and last 3 columns are nothing but the identity matrix. You can see over here diagonal elements are 1 and non diagonal elements are 0. So whatever the formula you are getting over here is the m1 first this row is multiplied with this column so m1 into 1 which is m1 plus m2 into 0 which is 0 plus m3 into 1 which is m3. So m1 plus m3 this is for u1 similarly for this second column m1 into 1 m1 plus m2 into 1 m2 plus m3 into 0 0. So likewise if you perform all the multiplication of rows through all the columns you are getting this final equation for u and this one is having a code bits of this code word u1 u2 u3 up to u6. So we know that these are the parity bits and these are the message bits. So finally you are getting the code word is what u1 u2 u3 u4 u6 using this method you can calculate all the code words of all data words 8 data words possible we seen. So for that 8 data words you are going to calculate all the 8 code words. These are the references. Thank you.