 We have been talking about BCH codes and how to think of code words of a BCH code, how to try to list out all the code words. We know one way of doing it with just the generator matrix and the parity check matrix, maybe we need a better idea, better way of going about it to simplify the implementation and all that. So that's where we are going. And the last point I was talking about was this observation that if c of x is the code word of the, say the T error correcting BCH code and let's say the block length n is 2 power m minus 1, all these things, I mean hopefully all these things you will assume from now on. So it's clear that it's primitive element. And I was talking about how we can see that x plus alpha times x plus alpha squared. So until x plus alpha part 2t has to divide c of x. So this is a code word if and only if this is true. So I was talking about understanding this properly, the main confusion was this is a polynomial in f2 power mx while I want this polynomial to be in f2x. So if you think of, I want to draw a picture here to illustrate what is it that we are trying to see. So you want c of x to be a factor of the polynomial here on the left, but you also want it to have binary coefficients. So if you think of this set, the set of all polynomials with coefficients, so this set is, if you picture this set as a big circle. So since the binary 0 and 1 are in f2 power m, where will f2x be? It will be inside it. I know that. So I can imagine this is my f2x. Now I want to imagine another set here which is the set of all multiples of this polynomial. Suppose all multiples of, so maybe I think I call it fx or something, they call it fx. So of this fx. Where will that set be in this? It will be some set here, but it is going to also intersect my f2x, am I right? So where will this bch code be in this? It will be this, exactly this. This will be my bch code, do you see this? So this is what we are trying to do. If you take all multiples of this fx, you will get all these polynomials which do not have binary coefficients also. I also want to impose this additional constraint, which means I will have to look at this intersection. Eventually what we will do is, in order to get a better description, eventually we will also describe this bch code as set of all multiples, when I say all multiples the degree needs to be a thing, multiples of some polynomial, some binary polynomial. Eventually we will get there, we will get there, we will see this, even this intersection is the set of all multiples of some binary polynomial. So to go from all multiples of fx to all multiples of binary polynomial, you need this notion of minimal polynomial. So you see that is the bridge between polynomial with coefficients from f2 power n and polynomials with coefficients from f2. So there was a question just before this lecture which disturbed me a little bit. See, you have to be careful when you think of, so I have been talking about polynomials in so many contexts. When I introduced finite fields, I talked about polynomials of degree less than or equal to m minus 1 and addition and multiplication modulo some irreducible polynomial. So we could think of the finite fields consisting of a set of polynomials. Maybe the polynomial alpha square plus alpha plus 1 is in the finite field. Now if I think of x square plus x plus 1 in f2x, is it the same as the alpha square plus alpha plus 1 in my finite field? It looks the same but it behaves very differently. Any multiplication I do in f2x, I do not have to do it modulo anything, I just multiply and I will be happy. But in my finite field, what am I doing? I am doing something else that alpha square plus alpha plus 1 in that finite field is different from the x square plus x plus 1. That is the reason why I shifted from x to alpha when I went to finite finite fields. I do not want this confusion. So when I talk about polynomials, do not think of them always as elements of some finite fields. Because you are doing addition and multiplication, modulo, some other, addition will not make any difference. Multiplication you are doing modulo, some irreducible polynomial. Once you do that, things behave completely differently. In fact, they have inverses. But a general polynomial does not have any inverse. You cannot multiply by some other polynomial to get 1. So they are completely different things. Do not get mixed up between the two. Do not think of one in terms of the other. When you think of polynomials, they are different. When you think of finite fields, they are different. Keep that in your mind, something important. So let us go back to these minimal polynomials. So let me remind you of what the definition was. I will write it, minimal polynomial of beta in F2M, least degree polynomial, least degree binary polynomial, I called it F beta x with beta as the root. That was my definition for the minimal polynomial. So you can go back now and imagine this set. So what am I, if you look at this set, F2Mx, this is my set of all polynomials with coefficients from F2 power m, there will be one polynomial which is x plus beta. What is the set of all polynomials which have beta as a root? All multiples of x plus beta, is that clear? All multiples of x plus beta are set of all polynomials with beta as a root in F2 power mx. So I will have that set. So this will be set of all multiples of x plus beta. In fact, this is the same as set of all polynomials which have beta as a root. In what, in F2 power mx? There is no problem here. So suppose if I were to ask, what is the least degree polynomial with beta as a root, what is the answer? X plus beta. Immediately you can get that. There is no problem. But what am I saying next? I am saying I do not want this. I want in the intersection of this guy and F2x, there will be some non-trivial intersection. What is the least degree polynomial in that? That is my F beta x. So you have to do some additional work to get there. So it is the same picture. You have to keep this picture in mind. When you think of polynomials over multiple fields, coefficients have to come from here and there. So this is the picture to keep in mind. Is that clear? So it turns out finding F beta of x is very trivial. It is very easy. That is the formula which you can directly substitute and find. So the observation there is, suppose I have F beta of x to be some, let us say, I do F0 plus F1x plus, so on till, let us say, Fr x to the power r. Suppose I say F beta of x is this. What do I know? Beta is the root of F beta of x and all these F0 to FR are what? Binary coefficients. So there are a few simple results you can derive. For instance, this polynomial cannot be reducible. What will happen if it is reducible? You can write it as a product of two other polynomials of lower degree and beta has to be root of one of them, which means immediately you will have a violation of the minimality of the degree. There was some other polynomial of lesser degree. So it is irreducible. So all these things you can easily prove. These are all properties. And there are other properties which you can prove. But the key property is once beta is a root and polynomials have binary, this polynomial has binary coefficient, what else should be a root? We know some things. Beta square has to be a root. Once beta gives you 0 when substituted here, I can happily square it. I know beta square will also be a root. Then what else will be a root? Beta power 4, beta power 8, so on. So just because beta was the root of this polynomial and just because it has binary coefficients, you know immediately beta square has to be a root, beta power 4 has to be a root, beta power 8, so on. So how far can you go? You cannot keep on repeating that and going on forever. Eventually you will fold back. So see what happens. So beta is a root of beta of x. So that implies, what else is a root? Beta square is a root, beta power 4 is a root, beta power 8 is a root. What will happen? Eventually when you keep on going, you will get beta power 2 power m minus 1. The m minus 1 is here, it is not 2 power m minus 1. You see the difference, right? 2 power m minus 1, minus 1 totally goes in the root. What will be the next power? Beta power 2 power m. But you know what that is? What is beta power 2 power m? It is equal to beta, right? Notice that what is beta power 2 power m? It is beta times beta power 2 power m minus 1 and what do we know about beta power 2 power m minus 1? It will become 1. So this becomes beta and you get the repetition. Yeah, yeah, so exactly. So I have to be careful here, okay? So it is not that this should be the first place where it repeats. But I know it will be a finite number only. I wrote that down carefully. It can repeat earlier than 2 power m minus 1 also. We will see some examples. But you know it will eventually repeat. It cannot keep on going forever, okay? It will eventually repeat and like you rightly pointed out, it can happen before m, okay? It can happen. Eminently, it can happen before m, okay? So we will see some examples to see how this repetition happens. But I know since beta is a root, all powers of beta raise to all 2 power, I mean not all powers of beta, only beta, beta square, beta power 4, so on, okay? So those powers should be powers of 2 by themselves, okay? So in anything, if you raise it to the power of 2, it should also be a root of f beta of x, okay? So let's see this in an example. Let's take the primitive element alpha and f8, okay? If you take the primitive element, okay? Let's try to find its, suppose f alpha of x is the minimal polynomial, okay? Suppose that is there. You know alpha is a root, then what? Alpha square is a root, and then alpha pa, 4 is a root, and then that's it, okay? These are also roots or roots of f alpha of x, okay? There could be other roots, I don't know. I know just because alpha was a root, these 3 are, these 2 other also are roots, okay? There could be other roots, but I don't know, okay? So this is how this works, okay? This is the first example. Second example I want to take is I'll start with alpha belonging to f16, which is primitive. Then I'll take beta to be, what shall we take it to be? We'll take it to be alpha power 5, okay? We'll take beta to be alpha power 5, okay? And then I want to look at f beta of x, okay? I want to look at roots of f beta of x, okay? Beta is a root, and what else is a root? Beta square is a root. What happens when you raise beta to the power 4? You come back to beta, okay? So you can only find beta and beta square as roots. You see that, right? When you raise beta square to the power 2 again, you get beta power 4, which is alpha power 20, which comes back to alpha power 5, okay? That clear? So only beta and beta square we're able to find. You can do other examples. So let's do a slightly fancy looking example. We'll go to f2, 256, primitive, and I'll look at alpha c, f alpha x, okay? What are its roots? We can say alpha, alpha squared, alpha power 4, alpha power 8, alpha power 16, alpha power 32, alpha power 64, alpha power 128. You know, for the primitive element, you will get all the m terms, right? For the primitive element, you have to get, because you know, the primitive element has got order equals 2 power m minus 1. So for the primitive element, you will get everything. But if the element is not primitive, then if the order is less than 2 power m minus 1, then something else can happen, okay? So if you try, for instance, if you take beta to be, what shall we take? Let's take... No, I'm not claiming anything like that, okay? There could be other roots, okay? So let me... If you want to be very specific, let me write that down. There could be other roots at this point. Eventually, we'll show that there are no other roots, but there could be other roots. At this point, there could be other roots, okay? But we will show... I mean, the next step, I'm going to show there are no other roots. So once you know that, it's very easy. So as I said, finding minimal polynomials is very easy in finite fields, okay? So if you take, for instance, what shall we take? I want a nice... Let's take alpha power 50, 51, yeah, 51, okay? I hope this will give me the answer I want, yeah, 51 we'll start with, okay? If beta of x, what can we conclude? Okay, you'll have beta, right? Then beta squared, which is what? Alpha power 102, okay? And then beta power 4, which is what? 204, okay? So alpha power 255 is 1, okay? Then next what? Do you get beta power 8? 153, okay? Will you get beta power 16? What's beta power 16? 51 is 17 times 3, 255 is 17 times 15, okay? So it'll come back, okay? Beta power 16 will become beta again, okay? So you see you get only four different roots in this case, okay? Okay? Well, there could be other roots, as you said at this point. We'll eventually see that there are no other roots, but there could be other roots. Likewise, you can see some interesting examples, another example is alpha power 85, okay? So if you do, let's say beta 1, okay? Then if you want to do beta 1x, you'll see beta, beta square, that's it, okay? Beta power 4 will become beta again, okay? 85 is 17 times 5 and 255 is 17 times 15, so we'll erase it to the power 4, it'll come back to beta, okay? So different kind of things like this can happen, but we know already that these have to be roots. There's no choice. Those have to be roots just because the polynomial had binary coefficient, okay? So in general, what can I say? If I have, if I have, if I have, if I want to generalize now, in general, okay? How many roots can we say it'll have? We have a beta, beta belonging to f2 power m, okay? And fbeta of x, okay? You could have, could have roots of fbeta of x being beta, beta squared, okay, right? So until, let's suppose some beta to the power 2 power r, okay? So maybe the next one became beta again, okay? So what am I saying when that happens? Beta power 2 power r plus 1 is what? Equal to? Equal to beta again, okay? So actually maybe I'll take r minus 1 here, I'm sorry, apologize for this. I'll take r minus 1 and r here, it's the same, okay? Beta power 2 power r, okay? So you can show, so, so looking at this very closely, you get beta power 2 power r minus 1 equals 1, okay? So this can happen, the smallest r for which this will happen, 2 power r minus 1 will have to divide, okay? So this is some careful number theory here, it's not very, it's not very difficult but one needs to think about it, there's some proof here, okay? 2 power r minus 1 will have to divide 2 power m minus 1, okay? Only then this can happen, okay? It's almost like the order of beta is 2 power r minus 1, right? I mean it's not really the order but you see what I mean, okay? This can only happen if this is, this is, this is true. From here you can even conclude r will have to divide m, okay? So 2 power r minus 1 will divide 2 power m minus 1, only when r divides, otherwise it won't divide, okay? You can prove these things but this is a useful fact, okay? So this can happen only when r divides m, okay? So let's keep that aside, I have not proved it but that's something you can look up to prove, it's not very difficult to prove this, okay? All right, so my claim now is going to be this f beta of x is equal to x plus beta, okay? This is our claim, x plus beta squared so on till x plus beta power 2 power r minus 1, okay? That's my claim. So you know this polynomial, right? Has to divide f beta of x that we know, right? Because all those things are roots of f beta of x. So that polynomial has to divide, that's why I kept saying there could be other roots, there could be other roots, maybe there could be other polynomials, irreducible factors. What I'm claiming here is there is no such thing, just multiply all these things, you get f beta of x, okay? So the way, there's a proper way of proving it but I'll just give you a simple intuitive argument for why it should be true, I think that's reasonably good proof also, what I'm going to give you but there is a proper way of proving it. So what do I need to prove now to show that this is equal to the minimal polynomial? Exactly, do you see that? If once I show all coefficients here are binary, then I'm done, okay? All I need to show that this is the minimal polynomial is that on the right-hand side here, x plus beta times x plus beta squared so on till x plus beta to the power 2 power r minus 1, all coefficients are binary, that's the only thing I have to show. Do you see that? Once I get that all coefficients are binary, that has to be the minimal polynomial, right? This polynomial has to divide it and all coefficients are binary, so it has to be the minimal polynomial, okay? So that's the proof that you need to show, okay? So the way to show that is how do I show any number is binary, okay? So what do I know in general? In general, this polynomial can belong to, in general it could have coefficients from, coefficients from which field? From where? F2 power m, right? Beta belongs to F2 power m, it could have coefficients from F2 power m, right? It multiplies sums and products from F2 power m, right? But what I'm actually saying is if you do all that, you will only get binary coefficients. It's a surprising result, it doesn't look like it very directly follows from there, right? But it's very surprising result that it has to be binary, okay? Okay? Is that clear? So to show this, the trick is, okay, we'll first prove it and then see a whole bunch of examples, okay? The trick in showing this is, suppose I give you an arbitrary element of F2 power m, okay? Some beta. How will you check if it's binary or not? What is the check? If it's the inverse of itself, okay? That's a good way. Another way of putting it is, if you square it, what should you get? It is the inverse of itself, yeah, but if you square it, what should you get? You should get the same, okay? But I don't know if it's the inverse of itself, I think that's not the, is that the logic? Yeah, for 0, then you can't use that logic, right? So the best check is if you square it, you should get the same thing, okay? Only for 0 and 1, that will happen, right? Beta square will be beta, only if beta is 0 or 1, just bring beta to the other side, factor beta out, you'll see beta is 0 or 1, you can't get anything else, okay? So beta square equals beta is the check, okay? So that's the same trick here. I will show if when you expand this and simplify, each coefficient when squared will give you the same answer, okay? So I'll give you a kind of a loose argument, it's not a proper way of doing it. The proper way of doing it is, you can find it in the books, okay? It's a nice way of doing it also, okay? Equals x plus beta, okay, so here's the sketch of the proof for the claim, x plus beta squared. So until x plus beta part 2 part r minus 1 equals, say, some f0 plus f1x plus, what will be the degree? Count, count carefully, let me think, should be r, okay? So fr x to the power r, in fact, fr can be, will be 1, okay? Right? Why is it r? See, you're going from 2 power 0, 2 power 1, all the way to 2 power r minus 1, that's why I adjusted this r minus 1 to r, okay, wanted to get, wanted to get degree r, okay? So you get this. So what will happen if I square both sides, okay? Here what will I get? If I square both sides, I'll get f1 squared x squared plus so on till fr squared x squared to the power r, okay, this is what I get. Let's say what will I get, okay, let's square term by term, I'm going to get x squared plus beta squared so on till what, right, you know, I'm sorry, if I square this, what will I get? x squared plus beta squared, if I square this, I'll get x squared plus beta squared. You see this, what I'm saying? I'm going to square the whole thing, which is squaring term by term. So if I square x plus beta, what do I get? x squared plus beta squared, what will be the square of x plus beta squared? x squared plus beta power 4, okay, so on till, what is the square of this? x squared plus beta to the power 2 power r, which is the same as beta, okay, beta to the power 2 power r is beta, okay? So look at these two polynomials, are they the same or different? They are the same except that I've replaced x with x squared and if I multiply and simplify, what will I get? Okay, what has happened here? These two polynomials, x goes to x squared, that's all, other than that it's the same, which means I can replace here each x with x squared and I would get the same thing, okay, I'm sorry, what happened? I think something happened here, okay, alright, right? So this guy is the same as f0 plus f1 x squared plus f2 x squared squared plus so on till fr x squared to the power r, this is equal to f0 squared plus f1 squared x squared, so on till fr squared x squared to the power r, do you see that? So now if you compare term by term, since this is equal always, you can compare term by term and see f i squared will have to be equal to f i, which means all the coefficients are binary, okay? So it's a little bit of a trickery to see how these terms occurred. If you're not very happy with this proof, I can tell you that there are other better proofs which just directly look at expressions for the coefficients and do it, okay? Did you have a question about this proof? Yeah, yeah, fr will be 1, yeah, it's okay, you can just keep it like that. Look at the left hand side expression, coefficient of x bar r is 1, okay? So that's my proof and there is a better proof using symmetric polynomials and proper arguments but this is also a good enough proof, I don't know, I'm happy with this proof when I see it, okay? Maybe people will not be very happy, okay? So let's see a whole bunch of examples to convince ourselves that this will indeed work the way we want it to work, okay? So I think, okay, so all these things will need f8, okay? And let's just do, I think f8 and f16 will need, it's gonna be difficult for me to produce f16 here, so maybe I'll, I think I have it in one of my previous lectures, I should probably put it in permanent memory so that I can pull it back one another I want, okay? So let's quickly write down f16, okay? So I'm gonna write down f16 as 0, 1, alpha, alpha squared, so until alpha power 14, I'll say alpha power 15 is 1 and then alpha power 4 is 1 plus alpha, okay? If I do this, let me write down the table for, let's say alpha power 3, alpha squared, alpha 1, okay, and 0 is gonna be 0, 0, 0, 0, 1 is gonna be 0, 0, 0, 1, alpha is gonna be 0, 0, 1, 0, 1, alpha squared is gonna be 0, 1, 0, 0, 1, alpha 3 is gonna be 0, 0, 0, 0, 4 is gonna be 0, 0, 1, 1, okay, alpha power 5 is gonna be 0, 1, 1, 0, 1, 0, 1, alpha power 6 is gonna be 1, 1, 0, 0, 0, let me know if I'm making a mistake. Alpha power 7 is gonna be 1, 0, 1, 1, am I right? Alpha power 8 is gonna be 0, 1, 0, 1, okay, alpha power 9 is gonna be 1, 0, 1, 0, alpha power 10 is gonna be 0, 1, 1, 1, alpha power 11 is gonna be 1, 1, 1, 0, alpha power 12 is gonna be, what do we get? 1, 1, 1, 1, alpha power 13 is gonna be 1, 1, 0, 1, alpha power 14 is gonna be 1, 0, 0, 1, there you go, think I've, let me just commit this to memory copy, okay, alright, so that's my f16, I wanna take f8, but f8 is so simple that you might be able to do that in a flash, okay, so I wanna do f16, I want you to go ahead and compute f alpha x, okay, it's some work but I want you to do it, okay, let's do f alpha x first, what's the formula for f alpha x, okay, first thing you have to figure out is that the list of roots starting from alpha, alpha, alpha square, alpha power 4 and alpha power 8, okay, and then you do this, x plus alpha times x plus alpha square times x plus alpha power 4 times x plus alpha power 8, okay, so if you simplify, you should get a binary polynomial which is the minimal polynomial of alpha, okay, so this is worth doing, spend your, spend a couple of minutes, okay, first best way of doing it is to first multiply these two guys, okay, you'll get x squared plus, what is alpha plus alpha squared, this is alpha power 5, okay, and then alpha power 3, then x squared plus alpha power 4 plus alpha power 8, give you 0, 1, 1, 0 which is alpha power 5 again, is that right, let me know if I make a mistake, okay, so then now we'll go ahead and do this entire computation, okay, so you'll get x power 3 term to vanish, and x square will also vanish, you'll get the x term and you'll get 1, right, the constant term is 1, everything worked out perfectly, okay, is it surprising that the minimal polynomial of alpha turned out to be x power 4 plus x plus 1, how many of you are surprised and how many of you are not surprised, okay, so look at that, look at the primitive polynomial with which we started, okay, x power 4 plus x plus 1 and we said alpha is going to be a root of that primitive polynomial, right, alpha is something which satisfies that primitive polynomial, alpha power 4 plus alpha plus 1 is 0, so obviously if you go back and compute the minimal polynomial of alpha, you should get x power 4 plus x plus 1, if you didn't get it then something was wrong, okay, so you see this will be very easy, for alpha it's very easy, okay, what about the minimal polynomial of alpha squared, let me see who's going to give me the smartest answer, oh, you don't have to do the computation, should be the same as this, why, alpha is the root of that and alpha squared is also a root of that and it's in a reducible polynomial, it'll be also the same thing, what about alpha power 4, okay, I've already put an equality, okay, so all these guys will be the same as what you already calculated, so you see finding all these guys will be equal to this, okay, it'll be equal to f alpha x, okay, you can find minimal polynomials by the dozens, okay, not by the dozens, by 4 at a time kind of, you know, you find 1, you found 4, okay, everything else comes for free, okay, so you don't have to redo the same computation, okay, so what's the next thing which we should try and find, alpha power 3, okay, go ahead and find it, okay, so you should have got x plus alpha power 3, x plus alpha power 6, x plus alpha power 12 and then x plus alpha power 9, right, okay, go ahead and multiply, simplify, okay, so you should get x power 4 plus x power 3 plus x squared plus x plus 1, okay, so you should get this, okay, so I mean don't waste your time now doing it but you will get it eventually, okay, so you will get x power 4 plus x power 3 plus x squared plus x plus 1, what's the next thing that I should try and find, what's the question, no, what's the doubt, please, yeah, how this polynomial became a radius, yeah, yeah, don't worry about that, okay, so next thing we have to find, okay, so alpha power 6 and 12 and 9 have already come for you, next thing to find is alpha power 5, okay, so you will see if you do this, it will be alpha power 5 and then alpha power 10, okay, the next thing would repeat and this will simplify as x squared plus x plus 1, okay, okay, the next thing to find is alpha power 7, okay, this will work out as x plus alpha power 7, okay, I've done this for a long time in my life, so I know all this by heart, okay, so if you're wondering how is it that I'm getting all this so fast because I know many of these things by heart, okay, so x power 4 plus x power 3 plus 1, okay, so there's lots of questions you can ask about the, I'm sorry, I have to find, no, no, no, no, no, no, well, I'm sorry, yeah, yeah, yeah, but you can get some interplace possibility, so those are the, those are the only things, after this you can stop because there will be no other element for which you have to find, minimal polynomials, everything is already there, okay, so there are lots of interesting things you notice, right, so for instance one point I made, once you have an irreducible polynomial, if you reverse the coefficients, right, order of the coefficients, you again get another irreducible polynomial, right, so x power 4 plus x plus 1 was irreducible and x power 4 plus x power 3 plus 1 is also irreducible, notice alpha is the root there and alpha power 14 is the root here and you'll see the inverse roots will hold, okay, alpha square is the root there, alpha power 13 is the root there, alpha power 4 is the root there, alpha power 11 is the root there, okay, it's all interesting the interplay between these things, so one can argue you don't even have to do this computation for alpha power 7, just by noticing that it's the inverse of that, you'll know what this is, you can easily write now, okay, so there are lots of interesting things like this which can, which will end up in quiz problems for instance, you know, so if you want to have enough practice you might be confused by these things, okay, there are lots of interesting properties in this computation, which can help you avoid a lot of drudgery and work, okay, so there are a lot of, a lot of deep results concerning minimal polynomials which we absolutely won't have any time to prove but I'll state it, okay, one of the things you can show which is very surprising is if you do this for f2 power m, okay, if you go ahead and find all the minimal polynomials of all elements of f2 power m, you will get all binary irreducible polynomials of degree r that divide m, okay, surprising, okay, so you do this f2 power m, do minimal polynomials for each of these elements like this, okay, you will get all the binary irreducible polynomials whose degree divides m, okay, for instance here what is m? 4, so I should get all binary irreducible polynomials of degree 1, 2 and 4, okay, how will I get degree 1 and 2? I should have done f0 and f1, if I done f0 I would get x, then for f1 I would have got x plus 1, okay, so I didn't do that, f0x is what? x itself and then f1x is x plus 1, okay, these two are very trivial, one doesn't have to do any problems, okay, what about degree 2? I should get all irreducible polynomials of degree 2, you see x2 plus x plus 1 shows up, okay, and I will get all irreducible for binary polynomials of degree 4, okay, those are the three, okay, so it's a fabulous result, okay, so you don't have to worry too much, sorry, you show that using properties, it's okay, I don't want to get into the proof, okay, if you want proof for all these things, there's a book on finite fields and applications by Lydl and Nader Rettem, okay, L-I-D-L, okay, that's the name of the first author, take a look at that book, finite fields and applications, okay, so that's about minimal polynomials, okay, so let's now go back to our, I can do this for other fields also, okay, so let's do it for other fields without having this table with us, okay, so let's do one more example, it's pretty interesting, suppose I now say I want to do this for let's say f64, okay, right, I don't know what this field is, maybe I don't even know what the minimal polynomial is, okay, so it's okay, but you'll see you can get amazing amount of information about what the minimal polynomial is without even knowing anything about the field, okay, it's just mirror arithmetic with 2 and 4 and all that, you'll see this is alpha is just a placeholder, okay, you can do everything with the exponent, it's nothing there about alpha which stopped you anywhere, okay, so what do I know, alpha per 63 is 1, okay, and I know alpha per 6 is something, okay, I don't even have to know that, okay, I don't even have to know that, you'll see I can get a lot of information about the minimal polynomials, okay, so what will be if I want to worry about f alpha x, okay, what will it be, okay, x plus alpha times x plus alpha squared times so on till x plus alpha to the power 32, right, it will have degree what, degree 6, okay, so what I can do is I'm just repeating this x plus x plus and brackets, I mean they don't have any meaning, okay, what really matters is what, the roots, what are the roots of f alpha of x, okay, so maybe we write alpha and then roots of f alpha of x here, okay, maybe we just do that, okay, so if we want to do that, if we have alpha then it would be alpha alpha squared, alpha power 4, alpha power 8, alpha power 16 and alpha power 32, okay, this contains essentially the same information as the top line, right, if I have to find f alpha of x what will I do, I'll do x plus alpha and then multiply, I don't care, okay, the next thing, the next thing I have to worry about is what, alpha power 3, right, I don't have to worry about alpha squared, okay, because I know already what its minimal polynomial roots will be, okay, so I'll do alpha power 3, what will I get, okay, we can do it, notice the only thing you need is alpha power 63 is 1, okay, I can happily keep on doing it, okay, alpha power 6, alpha power 12, alpha power 24, alpha power 48, then alpha power, well 96, 96 is the same as 33, okay, I can stop there, okay, then what is the next thing I have to worry about, alpha power 5, okay, so I would get alpha power 5, alpha power 10, alpha power 20, alpha power 40, alpha power, well 60, 60 is the same as 17 and then alpha power 34, okay, okay, do you see that, then what is the next thing I have to worry about, alpha power 7, okay, I would get alpha power 7, alpha power 14, alpha power 28, okay, then alpha power 56 and then alpha power 112 which is the same as what, I believe 49, am I right, yeah 49 and then alpha power 98 which is the same as 35, okay, so you notice here, I am again repeating this alpha power, alpha power, it has no meaning, okay, you see what I mean, right, I can convey the same information, we are completely dropping out that alpha, okay, for instance I can say what is the next power I am interested in, 9, okay, I can simply write 9, okay, you know it is alpha power 9, okay, I can simply write 9 and what will be the powers, what will be the roots of f alpha power 9x, it will be 9, what else will be there, 18, 36, 72 which is the same as 9, so I can stop there, okay, so it is enough if I provide this list, I do not have to keep writing this alpha power, alpha power, alpha power, why am I wasting ink, right, I do not have to do that, okay, it is enough if I simply write this, what is the next power I need to worry about, 10 is there already, 11, right, 11 would be 11, 22, 44, 88 is the same as 25, 50, 100 is the same as 37, okay, right, I can keep on doing this, next thing is 13 I believe, okay, so one can keep on doing this, okay, so maybe 13 I will do, just for fun, 26, 52, okay, 104 is the same as 41, 82 is the same as 19, okay and then 38, okay, so you see you can keep on doing this, it is very easy to get, what does this tell me, what is the degree of this, degree is 3, okay, all these guys have degree 6, these two also have degree 6, okay, so I can find the degree of the minimal polynomial with absolutely no other information about the field, okay, I do not need anything else, I can happily find the degree, okay, you can do this very simple work, okay, so then 15 is the next thing, okay, I would get 15, 30, 60, okay, then what do you get, I am sorry, 27, no, 27, no, not 27, I am sorry, 57, no, 120, 57 and 114 would be 51, am I right and then 102 would be 39, okay, is that right, yeah, so these are the degrees, I can keep on doing this, I do not want to exhaust this list, I can keep on doing this, okay, so, okay, okay, so, so, so notice given any power of alpha, how do I find the roots? Roots as powers of alpha, I have to take that power of alpha, then multiply it by 2, then multiply by 4, multiply by 8, so on till I get a repetition, okay, so if you form a set like that, this set is called something like cyclotomic coset generated by 15, okay, this is a technical name, I do not want to get into it, it is called some special name, okay, so it is some coset generated by that and there are lots of, you can study it very closely, but the essential idea is those are the roots of the minimal polynomial for alpha power 15, okay, one can always do this very, very easily, okay, so the next statement I want to leave you with before we come back and meet next week again is coming back to BCH codes, so what does all this mean for BCH codes? Let us go back to BCH codes, okay, what is it that we want, okay, so in BCH codes what did we have? C of x is a code word, code word of what? of a T error correcting, if somebody tells you this is true, then you know this implies this is if and only if what? x plus alpha, x plus alpha squared, so until what? x plus alpha part 2t divides C of x, okay, what is alpha now? Alpha is the primitive element in f2 power m, okay, so you know this is true, okay, now since you know C of x has binary coefficients, you can say a lot more, okay, so you know alpha, alpha squared to alpha power 2t or all, we will have to divide C of x, from here one can show this will happen if and only if LCM of f alpha x, f alpha squared x, f alpha power 3x, so on till f alpha power 2tx divides C of x, okay, so it is a very simple, it is easy to make this argument, okay, so because x plus alpha divides C of x, you know the and C of x has binary coefficients, okay, one can claim that the minimal polynomial of alpha has to divide C of x, okay, it is the same argument like I did before, you can prove it very easily, okay, alpha is a root, alpha squared is a root, alpha power 4 is a root, all those things are roots, and they do not have any common factors, so multiply them together, the minimal polynomial has to divide C of x, same thing I can do for f alpha squared, why? Exactly, all right, so people have foreseen what is going to happen in the future, but anyway, but why did I put the LCM there? Why did I say LCM here? Why can't I just say the product of all these things? Yeah, you have only the, so when you say each of those individually divide, only the LCM has to divide, right, because f alpha of x and f alpha squared of x are what? They are same polynomial, once you say that divides, you cannot again include f alpha squared of x also, so the LCM of this has to divide C of x, okay, what is so amazing now is, this is a polynomial in what? This belongs to f 2 power mx, where do you think this will belong? f 2 x, okay, and we have achieved what we wanted to achieve, okay, so that was our key problem, right, you remember that intersection, we had this problem of this polynomial having coefficients from f 2 power mx, we have now used the minimal polynomial idea and moved to a polynomial which has coefficients from f 2 x, and I know C of x is a codeword if and only if that divides, okay, this polynomial is called the generator polynomial of the BCH code, okay, so it's usually denoted g of x, okay, okay, so look at this powerful statement, C of x belongs to this BCH code if and only if C of x equals m of x times g of x, okay, and what is m of x now? m of x is any binary polynomial, can I say any binary polynomial, what should I say of degree such that what? Degree of C of x is less than or equal to n minus, okay, that's the only thing I have to worry about now, I know this g of x, I can take all possible binary polynomials m of x so that the product of those two should have degree less, so m of x itself should have degree less than or equal to something, okay, so we'll come back and look at that in the next week, okay, but this is what we have achieved, okay, so this is what we wanted to do, and we have achieved that using this idea of minimal polynomials to go from polynomials with coefficients from f 2 power m to polynomials with binary coefficient, okay.