Ore polynomials and applications to coding theory
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Caruso, Xavier
In the 1930’s, in the course of developing non-commutative algebra, Ore introduced a twisted version of polynomials in which the scalars do not commute with the variable. About fifty years later, Delsarte, Rothand Gabidulin realized (independently) that Ore polynomials could be used to define codes—nowadays calledGabidulin codes—exhibiting good properties with respect to the rank distance. More recently, Gabidulincodes have received much attention because of many promising applications to network coding, distributedstorage and cryptography. The first part of my talk will be devoted to review the classical construction of Gabidulin codes and presenta recent extension due to Martinez-Penas and Boucher (independently), offering similar performances butallowing for transmitting much longer messages in one shot. I will then revisit Martinez-Penas’ and Boucher’sconstructions and give to them a geometric flavour. Based on this, I will derive a geometric description ofduals of these codes and finally speculate on the existence of more general geometric Gabidulin codes.
 Thank you for the introduction. And thank you also for the invitation for giving me the opportunity to give this talk on ORA polynomial and application to coding theory. So let's start. So I first would like to introduce what are ORA polynomials. So this, can you see? Really? So it's setting in blue. It's better in black. Okay. Yeah. So what is? Oh. It's okay. So I'm going to write this as a, just copy the notation in the blackboard. So K is a field. So setting is the following. So K is a field. I consider morphism offering from K to K, which I call phi of phi. So morphism offering. And then I consider D from K to K, which is what we call a feed derivation. So what is, by definition, a feed derivation? It's an application from K to K, which is additive and satisfies this twisted lemnitz rule. So D of AB equals phi of A times D of B plus D of A times B. So it's what we call a feed derivation. And you see that if phi is the identity, then this condition is just the usual lemnitz rule. D of AB equals A times D of B plus D of A times B. Okay. And now I can define the O-ring, which I denote by K of X, phi and D. So it's a non-commutative ring in general. It is described as follows. So first I describe its elements. So elements are just usual univariate polynomials over K. So I insist that are univariate polynomials. So the variable is this X and this phi and this D just, in the notation, some extra letters for denoting the twisting is isomorphism. They are not variables. Then the addition is also very natural. It's a standard addition. But the multiplication is a bit different. It's a usual multiplication. It is twisted by this rule. If you have the variable X multiplied by a scalar on the right, A, then it's given by this expression, which is phi of A times X plus D of A. So we have something which is twisted. Okay. So this is the definition of O-ring polynomials. And let me show you some example of computation. So I recall you the rule here. And let me show you how we can compute X square times A. So you just write that it X times X A. And now use the commutation relation to say that X A is phi of A times X plus D of A. You expand everything. You get this expression. Then this X times phi of A, you can just reuse this rule with X replaced by phi of A in order to get an expression of this. Then you get this. X times D of A is something like that. There is not something important for the results. And then we refactor everything. And you get this expression for X square times A. So I think I have commutious that we can compute in the same way X cubed times A and X fourth times A and so on and so forth. And so we have an expression for X to Z times A for any integer N. Okay. And now if we want to compute the product of two O-ring polynomials, for instance two O-ring polynomials of Db2, like this, then you just expand everything without permitting any X and A, B, C, D. You get this expression. And now X cubed, X to the fourth, sorry, is X to the fourth. And the second sum X to the square times CX, you can rewrite it using the expression of X to the square times C. Get some here. And you get this expression. And so on. And you continue for the other summands. I don't want to be complete because the result is quite long. But you get so this computation is here for two, because I would like to, I wanted to convince you that this rule I used to define some multiplication, really define some multiplication in this ring of polynomials. Okay. And here another example. So if you are working over the complex number C, with, and you choose for the twisting isomorphism, the conjugacy, X maps to the conjugate of Z maps to the conjugate of Z. And if you are just taking zero for the derivation, so in what follows, I will often you take D equal zero and so I will omit it in notation. So in this, in this particular ring, then you have this computation, for instance, which is a remarkable identity. So X plus A times A times X minus the conjugate of A is X squared times the norm of A squared. Okay. So you can check this by just by expanding the, sorry, just by expanding the left hand side. And as a consequence, you see that our polynomials actually can admit many, many factorization, for instance, the polynomial X square minus one in this ring as factors as X plus A times X minus A conjugate for all A of norm one. If you can see many factorization for one polynomial, so it's a bit different from standard polynomials. But in any case, our polynomial look like usual polynomials. For several, several features with usual polynomials, for instance, the first thing is that we have a well-notion of degree, the well-defined notion of degree on our polynomial. So this is a usual standard degree. And this degree behave as we expect. So if you have two polynomial, or polynomials F and G, then the degree of the sum is at most the maximum of the degree of the summands. And the degree of the product is the sum of the degree of the terms. As you can imagine. Maybe something which is not so a little bit more unexpected is that this ring of all polynomials is right Euclidean. It admits right Euclidean division. So it's just this. If you have A and B two polynomials with B nonzero, then you can just write A equals QB plus R for Q and R two polynomials which are uniquely determined with this condition and the fact that the degree of the reminder is at most is less than, strictly less than the degree of the divisor. So standard fact accepts that you have to be careful that you really have to write A equals QB plus R and not A equals BQ plus R otherwise false. Anyway. And so the corollary is that this ring of all polynomials K of X, V, D is left principle, meaning that all left ideal or principle are narrated by one element. And you have a notion of right GCD and left LCR. So you have to be careful between left and right. But okay. So this property has really, so you can continue this list. For instance, there is a Euclidean, the usual Euclidean algorithm makes sense in this non-commutative context and so on and so on. Okay. So these are the polynomials. And I will of course come back on this, but this is a very standard property of them. I would just want to present in the first very short first part of my talk. And maybe you are wondering why we are interesting in this polynomials. So I will, in the rest of my talk I will explain some application to coding theory of this setting. But this polynomial was first introduced by O.A. in the 40s, I think, if I remember correctly. And the motivation was not coding theory at that time. It just one, two, it just because very, very standard construction in non-commutative algebra. It really helps. It's really the first example, maybe not the first, but one of the first examples of non-commutative algebra and non-commutative rings. It's just like polynomials are useful for studying rings. O.A. polynomials are really useful for studying non-commutative rings, non-commutating ring theory. It's a very central object. And it's also very useful for studying semi-linear algebra and linear differential equations. So for instance, if phi is the identity, then the ring of O.A. polynomials are nothing but the ring of linear differential operators. And so it's very useful for studying linear differential equations. And if d is zero, then it is related to semi-linear algebra, just like usual polynomials are related to linear algebra. So you have the characteristic polynomial factorization polynomial of random of theorem and so on. And if you are not interested in linear algebra, but semi-linear algebra for some reason, then the natural setting is this, or a polynomials. Okay. But it's not what I want to discuss in this talk. What I want to discuss is application to coding theory because, sorry, some. And so before I probably need to recall you some basics of coding theories. Although probably you will know what I'm, what would be on this slide. Anyway, so first of all, the definition of a linear code. So k is still a field. I gave a field as before. And so a linear code is a subspace, k linear subspace of k to the n for some integer n. Okay. So this integer n is called the length of the code. The dimension of the vector space is called the dimension of the code without any surprise. And another important notion on linear code is the minimal distance. So the minimal distance of a code is the minimal of w of x for any x in c is not zero. So the elements in c are usually called the code walls. So I will probably use this terminology in what follows, but I will try to avoid it. And w of x is what we call the aming weight. And the aming weight by definition of a vector. So x is an element of k to the n. So x1, x2, xn is a number of nonzero coordinates of x. So this defines a distance on k to the n. So the distance between x and y is a number, is a weight of the difference x minus one. And so it's also the number of coordinates on which x and y differ. So here is a beautiful example of linear code. So you can see, I hope, it's not so important to see this number here, but it's seven bits. So it's a natural subspace of f2 to the seven. So it's a linear code of f2 of dimension of length seven. Sorry. It has dimension three because one, two, three, four, five, six, seven, eight elements. And you can see that it's a linear subspace. You can check that it's a linear subspace. And it has a minimal distance four, I think, because each nonzero, actually each nonzero element of this code has four, one, and three, zero. So the minimal, actually it's even better. The weight of any nonzero element is four. So the minimal is also four, of course. Okay. And you have a theorem, a one-run theorem, which is called the singleton bound, saying that if you have any linear code, then there is a relation between the, I mean, inequality, inequality between the distance, the dimension, and the length, what is the minimal distance plus the dimension is, that's why it's at most the length plus one. So it's really easy to prove this theorem, this bound. And we might be interested in those codes for which this bound is actually an equality, as are called in general MDS code for maximal distance separable codes. Okay. So now I want to give you, sorry, I want to give you a very classical construction of codes, which are read solvent codes, which is very important for theory, in the theory. So the setting is the following. We have k and a, k and n, two positive integers with k less than n, or equal, but in general, k would be less than a, n. We are just considering a one, a two, a three, and so on, just until a n, pairwise distinct elements of k. So it's in place in particular that n is less than or equal to the continuity of k. So if you're working over finite fields, and this gives some restriction. And what is associated with salamone code is, I will denote it by read salamone of k and y and two and so on. It's the image of this mapping. So you take k of x, the ring of polynomial ring of k, usual polynomial ring, not twisted. This subscript less than k means that I'm considering all the polynomials of degree less than k, strictly less than k. And when I want, when I have such a polynomial, I can evaluate it at a one, a two, a three, and so on. And I get this way, a triple of n elements in k, and the code is by definition, all the tuples I can obtain this way from a polynomial of degree less than k. So of course, if k is n, then this map is a bijection. But if k is less than n, in general, it's only injective. And so I get a strict subspace of k to the n, which is called the read salamone code. And the theorem is that we really, we know the parameters of this code. So of course, the length is n by definition. It's not difficult to compute. The dimension is k because this map is injective. And the minimal distance is n minus k plus one. It's just, it's full of the fact that the polynomial of degree at most k minus one has at most k minus one zero. And so you see that in that particular case, singleton bound is reached, right? So it's an example of maximum distance parameter code. So this code is interesting for this reason. They are so interesting because they allow for an efficient decoding algorithm. So I don't want to speak about decoding in that talk. So I will skip this, but there are important codes. Okay. So that's for read salamone codes. And now I want to define what are, what are gabidun codes. And I will just show you that it's almost the same. I mean, it's a bit different, but it's the same spirit. So instead of taking a pairwise distinct element in k, I'm taking a linearly independent element of k. So linearly independent of a FP. And now I have to assume that k is a finite field FQ. So in that case, associated gabidun code, which is now denoted by gab of k and a one a n is the image of almost the same map except that I'm not considering all polynomial, but what we call linearized polynomial. And of height, in general, we say height instead of degree now, strictly says nk. And what is a linearized polynomial is something like this. Just a polynomial with a term in x, in x to the p, x to the p square, x to the p cube and so on, until x to the p to the k minus one. So it's called linearized because mapping x maps to x, x maps to x to the p, and so on, are FP linear. And so this is almost the same except that instead of evaluating polynomials of degree less than k, we are evaluating linearized polynomial of height less than k. Otherwise, exactly the same. And the point is that we have exactly the same theorem. The length, of course, is n, the dimension is still k, and the minimal distance is still l minus k plus one. So there are also maximum distance separable codes. And it's even better than k's because minimal distance is n plus k minus one, but for the having distance I've defined earlier, but also for another distance, which is finer in some sense, which is because the wrong distance, and what is the wrong distance? It's a kind of linearized version of aming distance. So instead of counting the number of coordinates of x, which are nonzero, we consider the span of all the coordinates of x, and we count the dimension of the span over FP. So if you have, I mean, if you have, for instance, if you have the vector one, one, one, one, one, then the dimension of the span is one, even if the coordinates are all different from zero. And so this wrong distance is less than the aming distance. It's not because of the aming distance, because it's rather clear. And so having a minimal distance, having the possible minimal distance for the wrong distance is better than having, I mean, yes, it's better than having the maximal minimal distance for the aming distance. So it's even better for this. Okay, so let me, so it was, so this code, so there are code gabidlin codes, but actually, that was first defined by del SART, and then rediscovered some years later by WOT, and then rediscovered a second time by gabidlin. But we are called, there are nowadays code gabidlin codes. So the first of all, they were described in this, in these terms, but now we prefer another way of thinking at them. Okay. Okay. So you see, I have this linearized polynomial, but the map, of course, the map x maps to x to the p is the formalus, x map to x to the p square is the square of the formalus. So in fact, I'm evaluating a polynomial in the formalus in a one, a two, a n. So instead of writing, sorry, instead of, so maybe it's not so, instead of, instead of writing f of a one, f of a two, f of a n, I prefer, I can just say that I'm evaluating a polynomial, I'm evaluating the polynomial f at the formalus and then apply it to a one, apply the map I get to a one a n. So now I no longer have a linearized polynomial, but just a standard polynomial, except that it's a polynomial in the formalus, and the point is that the formalus does not come into, does not come into a scalar. If you apply the formalus and then, sorry, no, if you multiply by a scalar and then apply the formalus, you assume that multiplying, applying the formalus, but multiplying by the scalar to the power p because of linearity. And so you have this commutation relation between the variable x and the scalar, x times a in this ring of polynomial is a to the p times x. And so it's exactly, I mean, we are, the domain is a, is a ORI ring, ORI ring and not a standard ring of polynomials. So, okay, so we can just make further simplifications instead of evaluating f of phi at a one, a two and so on. We can just give the f of phi, the linear mapping. So f of phi is a linear mapping. So he given it at, oh, sorry, yes. So I mean that I'm just mapping f of x to f of phi restricted to some subspace v, where v is a subspace generated by a one, a two, and a n, right? So now the image is not k to the n, but it's a set of homomorphism from v to k. And so instead of, and the rank distance now is a bit different. And so we don't have to compute the rank distance of some vector in k to the n, but so for some of some homomorphism from v to k, so the rank distance of u here is actually, if you, if you try to figure out what is this, this is the span of the core of the, the dimension of the span of the rows or the column of some matrix and just the rank of the matrix. So it's a reason for this one, one distance. Okay. And remember that a one, a two, up to a n, I just forgot it and replaced by the span of this. So instead of the data, the initial data is no longer a one, a n, but a vector space v, which is a, should be here, a sub vector space of k to the n and the gabidulin code I will denote just by gabidulin of k and v. So let me, let me maybe just read, read this once again. So I consider v a sub f p vector space of some f q and then the actuality gabidulin code, which is denoted by gabidulin of k and v is the image of this map mapping or a polynomial to this or a polynomial evaluated at the problem is restricted to v. If you don't want to restrict it, see what you can do also can take the equals case, perfect choice. Yeah. So what do we care about the wrong distance? So of course we can say that it's a very beautiful mathematical construction and that in, in fact, oh, I point out yours. No, sorry, the gabidulin code were defined maybe 50 years ago and indeed there were more or less forgotten, not exactly, but not, not that much, not such work has been done on it for some time and nowadays there is a much more attention on them. So because of practical applications, so there are a few of them, but the most standard one is application to what we call network coding. So network coding is, is this problematic. We have many computers distributed all over the world and this computer which is a server say and this one also has a message a and this one has a message b and this computer wants to transmit this message a to the computer here below and to this one. So two on the, on the bottom and the same for this computer, this server. So the message a should arrive here and there and the message b should arrive here and there as well. So what we can do of course is something like this for instance, you let the message a transmit, go to this pipe and the message a go to this, to this path for instance and the same for the message b, but you say that on the solution this, this pipe is used two times and so it's not so good, so good for the bandwidths. And you can just try to figure out what can be done, but in any, any case you have to use a pipe two times. But you can think at another solution which is the following. So I transmit a here and a and there and b here and there and so this computer has a, this computer receives a, this one receives b and this one receives a and b. And instead of transmitting a and b, it can just compute the sum of a and b, say a and b are elements in some field. I don't know, it's not so important. You can just view them as elements in some any vector space and so it computes a plus b and transmits the sum a plus b and then this one transmit a plus b on the left and then plus b on the right and so it means that this computer has received a and a plus b and so you can just by making the difference reconstruct b and the same on the right. Sorry. And so with this solution we see that we are, we are managed to transmit a and b to the both, we managed to transmit a and b to the both computers we, as we wanted to using only each pipe one time. So it's a solution which is more, more important using practice for network transmission and you see that the theory of natural, I mean distance as a classical coding theory is not really relevant in that case because I mean what we want to do, the problem we have to face is for instance if at some point some server is down or is infected by the various or is, I don't know, there is some problem with some server and in that case the whole message is corrupted. And so classical coding theory is not relevant but coding theory with respect to the wrong distance becomes relevant because if some compute, if for instance this message is altered then it will, eventually it will give an error of rank one and if two computers are, are, are, are, I don't know, are infected or something are down or I don't know, then, so resulting in a wrong, we have rank two and so on. And so because we are making all, lead our transformations. And so it's, it's a way to, to prevent, to, I mean to, to, to prevent communication in this model. So it's a reason why gabelian codes are important nowadays. Anyway, you, okay. So there is still another, no, okay. So there is still another generalization of gabelian codes that are related to present now, which is more recent. And for this I need to introduce more evaluation morphisms. So I have new assumptions. So for now I will assume that K is a field equipped with fee as before but K is no longer, needs, needs no longer to be a finite field. It can be any field. And I introduce this subfield F, which is a field of elements fixed by K. So F is a set of, a subfield of K consisting of elements such that F of X equal X. Right. And I assume that K over F, F is finite. So finite extension. So it's a Galois extension. Simply Galois extension with Galois group is generated by fee. In particular, it implies that fee has finite order. It's rejected. So as I said in the definition of gabelian codes, we have encoders, this evaluation morphism mapping F of X to F of phi. And the fact that we are working with our polynomial from the left ensures that this evaluation morphism is actually a morphism of rings. Commute with a multiplication. But actually, more generally, if we have any semilinear mapping from V to V, where V is a K vector space, so semilinear mapping is something which is additive and satisfies this action. So alpha of A X equals fee of A times alpha of X. So it's like linear mapping except that we have to twist the scalar by twisting morphism. So in that case, we have also well-defined evaluation morphism consisting in mapping an O-way polynomial F of X to F of alpha. Yeah. Sorry. Here the evaluation morphism of G of K or just K. The morphism of evaluation. And the morphism of K. So I don't have any question. Okay. So you can evaluate not only at fee, but at any alpha, which is semilinear mapping acting on some K vector space. And the proposition is that it's a classification proposition. It's rather straightforward to prove this proposition. But the semilinear mapping of K from K to K are just a multiplier of five. A times five for some A and K. And so using this, you see what you can do. Instead of evaluating only at fee, we can just evaluate at other alpha of this form and get this way of more general gabelian codes. So it's what we call the linearized gabelian codes, which were defined by Martin Espinaz recently. And also in a different spirit, with different notation by Delphine Boucher. But I will use the presentation of Martin Espinaz, which is more suitable for what I wanted to say here, today. So I consider as before K and M two integers. And A1, AM, elements of nonzero elements of K. And I assume that the norm of the AIs, the norm of K over A are pairwise distinct. So now the right condition. And I'm considering also V1, V2, Vm sub F vector spaces of K. So I'll play the role of the V as a previous slide. And then the linearized read salomon code, that's the city to this data, which is denoted by LRS, linearized read salomon of K, A1, V1, A2, V2, AM, Vm. It's the easy major of this mapping. So you take some array polynomial and we evaluate it at A1 phi, A2 phi, AM phi. So it's something which makes the gabelian construction and the read salomon construction. So now the natural points for evaluating array polynomials are these sublinear mappings. And I consider many of these sublinear mappings, A1, A1 phi, A1 phi, A2 phi, A3. And I evaluate the array polynomials at these points. Okay. And theorem is almost the same as before, is that the length of this code is, so it's a computation is a bit, the computation is, so we have to compute the dimension over K of the codomain of this map, which is a dimension of V1 plus a dimension of V2 plus a dimension of Vm. Sorry. Yeah. Yes, because A1 phi is a mapping from K to K. So the length is this. Yeah. So we can imagine that we are considering more general sublinear morphism and yeah, but I'm just sticking to this. So the length is n, the dimension is K, and again the minimal distance is n minus K plus 1. So say the minimal distance for the aming distance, but actually it's a minimal distance for refined distances before, which is called the same rank distance. So you have elements u1, um, ui lies is a morphism from Vi to K, and now you samed the rank of all this ui. Okay. So it's the notion of linearized read-salomon code, and I told you before that the gabin reading code were used for network coding and linearized read-salomon codes are also application in the concrete life and used for what we call the multi-shot linearized, multi-shot network coding. So I don't want to say much about this because I don't know that much actually, but it's not only a beautiful mathematical construction, it's also, it also has applications. Yeah. So now I would like to, maybe I can give this in the blackboard for a moment, and I would like to give you, to reinterpret this construction in a more general metric setting, spirit. So for this I need to give, to explain more things on orypoenomials. So let's first, let me first introduce this notation. R is the dimension of K over half. So it's the order of phi. And I consider this ring, which is the ring of Laurent orypoenomial, I don't know, so it's a orypoenomial in X and X minus one. And because phi is objective, it's not a problem because we have this commutation relation, which is X minus one times A is phi minus one of A times X minus one. So this ring has two important subrings. The first one is F to the X to the R and X to the minus R, Laurent polynomials in F in the variable X to the R. And it's actually the center of A. So it's quite, we'll play a quite important role, you can check this, it's not, it's not difficult. And in between, maybe I will write this again on the blackboard. So here you have A, which is K of X. Here you have the center and in between you have C, which is K of X to the R and to the minus R. So this is the center of A and C is the maximal sub-comutative sub-algebra. Okay. And what you can do is you can extend scalar to C from Z to C. And in that case on the, here, this way. So you have the tensor product. And since this C over Z is a Galois covering because it just obtained by extending the scalar from K to F, which is a Galois extension, then C times C, sorry, C tensor C over Z is just split as a product of copies of C, of R copies of C, C to the R. And here you have C tensor A and it turns out that C tensor A are also a very interesting interpretation. We have this isomorphism, C tensor A isomorphic to the algebra of C linear transformation of A. So the mapping is following. Okay. So C tensor A maps to the map, C acts by multiplication on the left and A acts by multiplication on the right. And the important point is that A is a C module, of course, but which is free of rank R. The basis, for instance, is 1 X, X square, X to the cube and X to the R minus 1. And so this is just a matrix algebra over C of dimension with R rows and R columns. So A is some or a polynomial, but when we extend scalar from Z to C, then we get a matrix algebra. And if you know algebraic geometry, Mike knows that this is a definition, almost a definition of Azumaya algebra. So Azumaya algebra is an algebra, we become isomorphic to a matrix algebra after an et al extension. And here's an extension C over Z is et al. Because it just, it comes from, as I said before, it's a Galois covering, it comes from the K over, it's deduced by scalar extension from the extension K over F, which is Galois. So we have an Azumaya algebra and when you have an Azumaya algebra, you have a lot of interesting facts. So for instance, we have what we call the radius trace coming from the trace. So when you have a matrix algebra, you have the trace determinant and so on. And this descend to A. So you have a radius trace, a radius trace map, which comes from A to Z, I don't know, which is given by this explicit formula. It's not so important if you can see the formula, just to say that it's explicit. And in the same way, you have the radius norm and the radius norm corresponding to the determinant. And also you have an explicit formula only for polynomial of degree one, otherwise much more complicated, but anyway, I won't show them. And now I just rephrase this. So it's a commutative variable I had before. And now I can just make a picture instead of diagrams. So I draw the spectrum of the center, spectrum of the, just a straight line. It's A1 with 1.3 moved. I have a covering of spec Z by spec C, which is a straight line, but defined over K, not over F. I have another copy of spec C here, and now I have the tensor product. So it's a fiber product and is a Dijon student of R copies of spec C, so R equals 2 in my drawing. And so above one point, I have two points. And I can consider the vector space, the K vector space generated by these two points, which are drawn here. And when I let the point vary on the axis, on the axis, I can just make the construction vary on the, so the vector space move on the top. And what I get is a vector bundle. And this isomorphism, saying that C tensor by A over Z is equal to some matrix algebra, just says that C tensor A is the under morphism of this vector bundle. And in fact, we can just, you can just let phi act on the under morphism of this vector bundle, and so A can be recovered under morphism of this vector model stable by phi. So I don't know how to draw spec A because A is non-commutative and I'm not really, I don't really know how to draw the spectrum of a non-commutative algebra, so I just don't do it, but I'm doing the under morphism of the vector bundle. Can you say again what the vector bundle, what the fibers, it's a vector bundle over C? It's a vector bundle over C whose fiber is just, so it's a trivial vector bundle over C of dimension R. And if I have some point Z on the, on spec Z, I can try to figure out what is the fiber of A over Z, which is by definition this quotient. And by the general theory of azimaya algebra, you know that it's a simple central algebra over Z. The point is that this vector bundle does not descend to C. And so it's a simple central algebra, which is in general not under morphism of algebra, just a simple central algebra, but for instance, if you are working over a finite field, all simple central algebra are matrix algebra. And in fact, we have characterization. This simple central algebra is split if and only if Z is a norm in this extension. In that case, yeah, so in that case, it means that if it splits, it's under morphism of some vector space over Z. And so I'm just drawing this vector space here, but you have to be very careful. This is the vector bundle that doesn't descend on Z, so just the fiber here. And so in that case, this isomorphism, I mean the splitting of this isomorphism is given by this map, this map evaluation of A phi I've defined before. So this evaluation of mapping f of X to f of A phi, realize this splitting. And so it's a way to see that the fiber of A over Z acting on this vector space. And so what is linearized with Salomon code, you just have some element of A, and each time you have a point on spec Z, you can evaluate it. You have this evaluation map, this evaluation map gives you an an anamorphism of K, and K can be seen as something laying above Z. So what you have to maybe to retain is that the gabidrin code actually is something laying over, leaving over a point, the point one, we have a vector bundle of dimension, vector space of dimension one over a point, and the linearized gabidrin code is the same situation, but we just evaluate that not only at one, but at many other points. Okay so it's a drawing. And you can wonder if we can do this in a more general situation, instead of having a covering of P1 by P1 or A1 with one point removed by A1 with one point removed, can try to do this in a more geometric situation, we have a curve on the bottom and another covering which is a cyclic Galois covering. But it's actually a working progress, I won't say so much about this, but it's something which would be linearized geometric codes or something like this. So it's a very, very early stage, so I just started this work with Amore Drouin, we will be a PhD student of mine next year. So in the five, ten minutes I have, I will just want to speak about duality. Maybe I won't finish what I want to say about this, but I think it's important. So the starting point is this theorem. If you have a read salomon code, which was defined, I remember by this evaluation formula in the very classical setting, then we can define the dual of this code. So by definition the dual is the orthogonal for the usual standard product of K to the N, on K to the N, natural pairing. And the dual of this code is the image of, can be described as the image of an anothomapping, given in terms of residues. This is quite interesting because the dual of a code in general is defined, is an intersection of hyperplane, but it is described as an image of something else. The proof of this is not difficult, it is a residue formula and an argument dimension. But there is an interesting corollary that using this description you can prove that the dual of read salomon codes actually also meet the single ton bound. So not only the read salomon codes are MDS, but the dual of them are also MDS. And what I want to do is try to generalize this result to a linearized read salomon code. So if I have the dual of a linearized read salomon code, which is this code I defined before, I want to have a description of the image of the dual. So there are some difficulties. So of course we are now looking for the image of something coming from, defined in terms of orypoenomials. The problem is that we don't know what is the residue for orypoenomials, we have to define this. And the second point is that now we no longer have something with the image in K to the end, but in this andomorphism of K to the M, and so we have to define a pairing on this space. So there are two points to do, two things to do. So I'm starting with a residue. For this I need to figure out what is a good orypoenomials. So the expansion is nothing but trying to evaluate orypoenomials on some thickening. And so I say orypoenomials, andomorphism of some vector models. The point is that on the left hand side, of course, this is a trigger vector model, you can just consider the image of a thickening of a point, and then you get a thickening of the vector models, no problem. But on the right side, it's not so clear what you get. And here comes a theorem saying that if you have a close point in spexies, that is an orypoenomial in Z, then there is an isomorphism between this projective limit corresponding to the function of the thickening and a ring of power series ring over the special fiber. And okay, maps A and 2H. And from this we can define a residue of orypoenomials. So first of all I need to define the fraction field of A, because if I want a residue I want a fraction. So it's defined as A tensor of fractions, quotient. So okay, I skip this. And now I have the residue map, so I'm considering N a reducible polynomial in Z. If I have a orypoenomial I can just map it to it, consider this Taylor expansion. And I can use the previous theorem to see the Taylor expansion as a real, I mean as a concrete series in H. And I can just consider the fraction field on the left, then it's since N maps to H, it maps to this projective limit where N is inverted. And on the right hand side it maps to a low run series with coefficient in A over N A. And now I can consider the coefficient in H minus 1 as for the usual residues. And I call this composite a residue at N. And I can define the residue at A as the residue at this polynomial which is the point Z over which A is defined and I evaluate at A is this. And this way I get an endomorphism of K. Okay, yeah, just a minute. So I just want one slide. And now I can define the pairing of the endomorphism of A for this I need to proceed by steps. Okay, so I first define a pairing of K which is a usual pairing given by the trace. Sorry. Then I can define a pairing of the endomorphism of K by taking this usual formula, the trace of the I joint of U composed with V. And then I have a pairing on the mapping which is the sum of the trace. Okay. And this can actually be put back on our polynomials. So the pairing is defined by this formula. You see that the trace and the I joint and the turn out that the trace and the I joint have some analogs on our polynomials. So for the trace we have seen that we have a radius trace here and it's easy to prove that the radius says it makes this I and I commute so it's when you evaluate. So after evaluation the radius trace becomes a trace. And it already himself define a pairing on our polynomials which is defined by this formula. So we define a junction on our polynomial defined by this formula. Sum of A I X I is the I joint of this is the sum of X minus Y A I. And after evaluation this becomes a natural pairing. This adjuncts becomes the I joint I defined before on the anamorphism of K. And using this you see that the pairing of evaluation of G and evaluation of F at this point is actually the radius trace of G star F evaluated at some point. And you can yes, yes, yes, yes, finished. And you have the same thing for the residue. The pairing of the residue and evaluation is the residue of the radius trace of something. And using this you can finally prove this theorem which is that the dual of the linearized with Salomon code is the image of this, of a code like this, where you have the residue at A I inverse of something like this. And the corollary of this is that the dual of linearized with Salomon code means a singleton bond as well as for with Salomon codes. And I'm done. Thank you.
 Thank you for the introduction.  And thank you also for the invitation for giving me the opportunity to give this talk  on ORA polynomial and application to coding theory. So let's start. So I first would like  to introduce what are ORA polynomials. So this, can you see? Really? So it's setting  in blue. It's better in black. Okay.  Yeah. So what is? Oh. It's okay. So I'm going to write this as a,  just copy the notation in the blackboard. So K is a field. So setting is the following.  So K is a field. I consider morphism offering from K to K, which I call phi of phi. So morphism  offering. And then I consider D from K to K, which is what we call a feed derivation.  So what is, by definition, a feed derivation? It's an application from K to K, which is  additive and satisfies this twisted lemnitz rule. So D of AB equals phi of A times D of  B plus D of A times B. So it's what we call a feed derivation. And you see that if phi  is the identity, then this condition is just the usual lemnitz rule. D of AB equals A times  D of B plus D of A times B. Okay. And now I can define the O-ring, which I denote by  K of X, phi and D. So it's a non-commutative ring in general. It is described as follows.  So first I describe its elements. So elements are just usual univariate polynomials over  K. So I insist that are univariate polynomials. So the variable is this X and this phi and  this D just, in the notation, some extra letters for denoting the twisting is isomorphism.  They are not variables. Then the addition is also very natural. It's a standard addition.  But the multiplication is a bit different. It's a usual multiplication. It is twisted  by this rule. If you have the variable X multiplied by a scalar on the right, A, then it's given  by this expression, which is phi of A times X plus D of A. So we have something which  is twisted. Okay. So this is the definition of O-ring polynomials. And let me show you  some example of computation. So I recall you the rule here. And let me show you how we  can compute X square times A. So you just write that it X times X A. And now use the  commutation relation to say that X A is phi of A times X plus D of A. You expand everything.  You get this expression. Then this X times phi of A, you can just reuse this rule with  X replaced by phi of A in order to get an expression of this. Then you get this. X times  D of A is something like that. There is not something important for the results. And then  we refactor everything. And you get this expression for X square times A. So I think I have commutious  that we can compute in the same way X cubed times A and X fourth times A and so on and  so forth. And so we have an expression for X to Z times A for any integer N. Okay. And  now if we want to compute the product of two O-ring polynomials, for instance two O-ring  polynomials of Db2, like this, then you just expand everything without permitting any X  and A, B, C, D. You get this expression. And now X cubed, X to the fourth, sorry, is X  to the fourth. And the second sum X to the square times CX, you can rewrite it using  the expression of X to the square times C. Get some here. And you get this expression.  And so on. And you continue for the other summands. I don't want to be complete because  the result is quite long. But you get so this computation is here for two, because I would  like to, I wanted to convince you that this rule I used to define some multiplication,  really define some multiplication in this ring of polynomials. Okay. And here another  example. So if you are working over the complex number C, with, and you choose for the twisting  isomorphism, the conjugacy, X maps to the conjugate of Z maps to the conjugate of Z. And if you  are just taking zero for the derivation, so in what follows, I will often you take D equal  zero and so I will omit it in notation. So in this, in this particular ring, then you  have this computation, for instance, which is a remarkable identity. So X plus A times  A times X minus the conjugate of A is X squared times the norm of A squared. Okay. So you  can check this by just by expanding the, sorry, just by expanding the left hand side. And  as a consequence, you see that our polynomials actually can admit many, many factorization,  for instance, the polynomial X square minus one in this ring as factors as X plus A times  X minus A conjugate for all A of norm one. If you can see many factorization for one  polynomial, so it's a bit different from standard polynomials. But in any case, our polynomial  look like usual polynomials. For several, several features with usual polynomials, for  instance, the first thing is that we have a well-notion of degree, the well-defined  notion of degree on our polynomial. So this is a usual standard degree. And this degree  behave as we expect. So if you have two polynomial, or polynomials F and G, then the degree of  the sum is at most the maximum of the degree of the summands. And the degree of the product  is the sum of the degree of the terms. As you can imagine. Maybe something which is  not so a little bit more unexpected is that this ring of all polynomials is right Euclidean.  It admits right Euclidean division. So it's just this. If you have A and B two polynomials  with B nonzero, then you can just write A equals QB plus R for Q and R two polynomials  which are uniquely determined with this condition and the fact that the degree of the reminder  is at most is less than, strictly less than the degree of the divisor. So standard fact  accepts that you have to be careful that you really have to write A equals QB plus R and  not A equals BQ plus R otherwise false. Anyway. And so the corollary is that this ring of  all polynomials K of X, V, D is left principle, meaning that all left ideal or principle are  narrated by one element. And you have a notion of right GCD and left LCR. So you have to  be careful between left and right. But okay. So this property has really, so you can continue  this list. For instance, there is a Euclidean, the usual Euclidean algorithm makes sense  in this non-commutative context and so on and so on. Okay. So these are the polynomials.  And I will of course come back on this, but this is a very standard property of them. I  would just want to present in the first very short first part of my talk. And maybe you  are wondering why we are interesting in this polynomials. So I will, in the rest of my  talk I will explain some application to coding theory of this setting. But this polynomial  was first introduced by O.A. in the 40s, I think, if I remember correctly. And the motivation  was not coding theory at that time. It just one, two, it just because very, very standard  construction in non-commutative algebra. It really helps. It's really the first example,  maybe not the first, but one of the first examples of non-commutative algebra and non-commutative  rings. It's just like polynomials are useful for studying rings. O.A. polynomials are really  useful for studying non-commutative rings, non-commutating ring theory. It's a very  central object. And it's also very useful for studying semi-linear algebra and linear  differential equations. So for instance, if phi is the identity, then the ring of O.A.  polynomials are nothing but the ring of linear differential operators. And so it's very useful  for studying linear differential equations. And if d is zero, then it is related to semi-linear  algebra, just like usual polynomials are related to linear algebra. So you have the characteristic  polynomial factorization polynomial of random of theorem and so on. And if you are not interested  in linear algebra, but semi-linear algebra for some reason, then the natural setting is  this, or a polynomials. Okay. But it's not what I want to discuss in this talk. What  I want to discuss is application to coding theory because, sorry, some. And so before  I probably need to recall you some basics of coding theories. Although probably you  will know what I'm, what would be on this slide. Anyway, so first of all, the definition  of a linear code. So k is still a field. I gave a field as before. And so a linear code  is a subspace, k linear subspace of k to the n for some integer n. Okay. So this integer  n is called the length of the code. The dimension of the vector space is called the dimension  of the code without any surprise. And another important notion on linear code is the minimal  distance. So the minimal distance of a code is the minimal of w of x for any x in c is  not zero. So the elements in c are usually called the code walls. So I will probably  use this terminology in what follows, but I will try to avoid it. And w of x is what  we call the aming weight. And the aming weight by definition of a vector. So x is an element  of k to the n. So x1, x2, xn is a number of nonzero coordinates of x. So this defines  a distance on k to the n. So the distance between x and y is a number, is a weight of  the difference x minus one. And so it's also the number of coordinates on which x and y  differ. So here is a beautiful example of linear code. So you can see, I hope, it's not so  important to see this number here, but it's seven bits. So it's a natural subspace of  f2 to the seven. So it's a linear code of f2 of dimension of length seven. Sorry. It  has dimension three because one, two, three, four, five, six, seven, eight elements. And  you can see that it's a linear subspace. You can check that it's a linear subspace. And  it has a minimal distance four, I think, because each nonzero, actually each nonzero element  of this code has four, one, and three, zero. So the minimal, actually it's even better.  The weight of any nonzero element is four. So the minimal is also four, of course. Okay.  And you have a theorem, a one-run theorem, which is called the singleton bound, saying  that if you have any linear code, then there is a relation between the, I mean, inequality,  inequality between the distance, the dimension, and the length, what is the minimal distance  plus the dimension is, that's why it's at most the length plus one. So it's really easy  to prove this theorem, this bound. And we might be interested in those codes for which  this bound is actually an equality, as are called in general MDS code for maximal distance  separable codes. Okay. So now I want to give you, sorry, I want to give you a very classical  construction of codes, which are read solvent codes, which is very important for theory,  in the theory. So the setting is the following. We have k and a, k and n, two positive integers  with k less than n, or equal, but in general, k would be less than a, n. We are just considering  a one, a two, a three, and so on, just until a n, pairwise distinct elements of k. So it's  in place in particular that n is less than or equal to the continuity of k. So if you're  working over finite fields, and this gives some restriction. And what is associated with  salamone code is, I will denote it by read salamone of k and y and two and so on. It's  the image of this mapping. So you take k of x, the ring of polynomial ring of k, usual  polynomial ring, not twisted. This subscript less than k means that I'm considering all  the polynomials of degree less than k, strictly less than k. And when I want, when I have  such a polynomial, I can evaluate it at a one, a two, a three, and so on. And I get  this way, a triple of n elements in k, and the code is by definition, all the tuples  I can obtain this way from a polynomial of degree less than k. So of course, if k is  n, then this map is a bijection. But if k is less than n, in general, it's only injective.  And so I get a strict subspace of k to the n, which is called the read salamone code.  And the theorem is that we really, we know the parameters of this code. So of course,  the length is n by definition. It's not difficult to compute. The dimension is k because this  map is injective. And the minimal distance is n minus k plus one. It's just, it's full  of the fact that the polynomial of degree at most k minus one has at most k minus one  zero. And so you see that in that particular case, singleton bound is reached, right? So  it's an example of maximum distance parameter code. So this code is interesting for this  reason. They are so interesting because they allow for an efficient decoding algorithm.  So I don't want to speak about decoding in that talk. So I will skip this, but there  are important codes. Okay. So that's for read salamone codes. And now I want to define  what are, what are gabidun codes. And I will just show you that it's almost the same. I  mean, it's a bit different, but it's the same spirit. So instead of taking a pairwise distinct  element in k, I'm taking a linearly independent element of k. So linearly independent of a  FP. And now I have to assume that k is a finite field FQ. So in that case, associated gabidun  code, which is now denoted by gab of k and a one a n is the image of almost the same  map except that I'm not considering all polynomial, but what we call linearized polynomial. And  of height, in general, we say height instead of degree now, strictly says nk. And what  is a linearized polynomial is something like this. Just a polynomial with a term in x,  in x to the p, x to the p square, x to the p cube and so on, until x to the p to the  k minus one. So it's called linearized because mapping x maps to x, x maps to x to the p,  and so on, are FP linear. And so this is almost the same except that instead of evaluating  polynomials of degree less than k, we are evaluating linearized polynomial of height  less than k. Otherwise, exactly the same. And the point is that we have exactly the  same theorem. The length, of course, is n, the dimension is still k, and the minimal  distance is still l minus k plus one. So there are also maximum distance separable codes.  And it's even better than k's because minimal distance is n plus k minus one, but for the  having distance I've defined earlier, but also for another distance, which is finer  in some sense, which is because the wrong distance, and what is the wrong distance? It's  a kind of linearized version of aming distance. So instead of counting the number of coordinates  of x, which are nonzero, we consider the span of all the coordinates of x, and we count  the dimension of the span over FP. So if you have, I mean, if you have, for instance,  if you have the vector one, one, one, one, one, then the dimension of the span is one,  even if the coordinates are all different from zero. And so this wrong distance is less  than the aming distance. It's not because of the aming distance, because it's rather  clear. And so having a minimal distance, having the possible minimal distance for the wrong  distance is better than having, I mean, yes, it's better than having the maximal minimal  distance for the aming distance. So it's even better for this. Okay, so let me, so it was,  so this code, so there are code gabidlin codes, but actually, that was first defined  by del SART, and then rediscovered some years later by WOT, and then rediscovered a second  time by gabidlin. But we are called, there are nowadays code gabidlin codes. So the first  of all, they were described in this, in these terms, but now we prefer another way of thinking  at them. Okay. Okay. So you see, I have this linearized polynomial, but the map, of course,  the map x maps to x to the p is the formalus, x map to x to the p square is the square of  the formalus. So in fact, I'm evaluating a polynomial in the formalus in a one, a two,  a n. So instead of writing, sorry, instead of, so maybe it's not so, instead of, instead  of writing f of a one, f of a two, f of a n, I prefer, I can just say that I'm evaluating  a polynomial, I'm evaluating the polynomial f at the formalus and then apply it to a one,  apply the map I get to a one a n. So now I no longer have a linearized polynomial, but  just a standard polynomial, except that it's a polynomial in the formalus, and the point  is that the formalus does not come into, does not come into a scalar. If you apply the formalus  and then, sorry, no, if you multiply by a scalar and then apply the formalus, you assume  that multiplying, applying the formalus, but multiplying by the scalar to the power p because  of linearity. And so you have this commutation relation between the variable x and the scalar,  x times a in this ring of polynomial is a to the p times x. And so it's exactly, I  mean, we are, the domain is a, is a ORI ring, ORI ring and not a standard ring of polynomials.  So, okay, so we can just make further simplifications instead of evaluating f of phi at a one, a  two and so on. We can just give the f of phi, the linear mapping. So f of phi is a linear  mapping. So he given it at, oh, sorry, yes. So I mean that I'm just mapping f of x to  f of phi restricted to some subspace v, where v is a subspace generated by a one, a two,  and a n, right? So now the image is not k to the n, but it's a set of homomorphism  from v to k. And so instead of, and the rank distance now is a bit different. And so we  don't have to compute the rank distance of some vector in k to the n, but so for some  of some homomorphism from v to k, so the rank distance of u here is actually, if you, if  you try to figure out what is this, this is the span of the core of the, the dimension  of the span of the rows or the column of some matrix and just the rank of the matrix. So  it's a reason for this one, one distance. Okay. And remember that a one, a two, up to  a n, I just forgot it and replaced by the span of this. So instead of the data, the  initial data is no longer a one, a n, but a vector space v, which is a, should be here,  a sub vector space of k to the n and the gabidulin code I will denote just by gabidulin  of k and v. So let me, let me maybe just read, read this once again. So I consider v a sub  f p vector space of some f q and then the actuality gabidulin code, which is denoted  by gabidulin of k and v is the image of this map mapping or a polynomial to this or a polynomial  evaluated at the problem is restricted to v. If you don't want to restrict it, see what  you can do also can take the equals case, perfect choice. Yeah. So what do we care about  the wrong distance? So of course we can say that it's a very beautiful mathematical construction  and that in, in fact, oh, I point out yours. No, sorry, the gabidulin code were defined  maybe 50 years ago and indeed there were more or less forgotten, not exactly, but not, not  that much, not such work has been done on it for some time and nowadays there is a much  more attention on them. So because of practical applications, so there are a few of them,  but the most standard one is application to what we call network coding. So network coding  is, is this problematic. We have many computers distributed all over the world and this computer  which is a server say and this one also has a message a and this one has a message b and  this computer wants to transmit this message a to the computer here below and to this one.  So two on the, on the bottom and the same for this computer, this server. So the message  a should arrive here and there and the message b should arrive here and there as well. So  what we can do of course is something like this for instance, you let the message a transmit,  go to this pipe and the message a go to this, to this path for instance and the same for  the message b, but you say that on the solution this, this pipe is used two times and so it's  not so good, so good for the bandwidths. And you can just try to figure out what can be  done, but in any, any case you have to use a pipe two times. But you can think at another  solution which is the following. So I transmit a here and a and there and b here and there  and so this computer has a, this computer receives a, this one receives b and this one  receives a and b. And instead of transmitting a and b, it can just compute the sum of a  and b, say a and b are elements in some field. I don't know, it's not so important. You can  just view them as elements in some any vector space and so it computes a plus b and transmits  the sum a plus b and then this one transmit a plus b on the left and then plus b on the  right and so it means that this computer has received a and a plus b and so you can just  by making the difference reconstruct b and the same on the right. Sorry. And so with  this solution we see that we are, we are managed to transmit a and b to the both, we managed  to transmit a and b to the both computers we, as we wanted to using only each pipe one  time. So it's a solution which is more, more important using practice for network transmission  and you see that the theory of natural, I mean distance as a classical coding theory is not  really relevant in that case because I mean what we want to do, the problem we have to  face is for instance if at some point some server is down or is infected by the various  or is, I don't know, there is some problem with some server and in that case the whole  message is corrupted. And so classical coding theory is not relevant but coding theory with  respect to the wrong distance becomes relevant because if some compute, if for instance this  message is altered then it will, eventually it will give an error of rank one and if two  computers are, are, are, are, I don't know, are infected or something are down or I don't  know, then, so resulting in a wrong, we have rank two and so on. And so because we are  making all, lead our transformations. And so it's, it's a way to, to prevent, to, I  mean to, to, to prevent communication in this model. So it's a reason why gabelian codes  are important nowadays. Anyway, you, okay. So there is still another, no, okay. So there  is still another generalization of gabelian codes that are related to present now, which  is more recent. And for this I need to introduce more evaluation morphisms. So I have new assumptions.  So for now I will assume that K is a field equipped with fee as before but K is no longer,  needs, needs no longer to be a finite field. It can be any field. And I introduce this  subfield F, which is a field of elements fixed by K. So F is a set of, a subfield of K consisting  of elements such that F of X equal X. Right. And I assume that K over F, F is finite. So  finite extension. So it's a Galois extension. Simply Galois extension with Galois group is  generated by fee. In particular, it implies that fee has finite order. It's rejected.  So as I said in the definition of gabelian codes, we have encoders, this evaluation morphism  mapping F of X to F of phi. And the fact that we are working with our polynomial from the  left ensures that this evaluation morphism is actually a morphism of rings. Commute  with a multiplication. But actually, more generally, if we have any semilinear mapping  from V to V, where V is a K vector space, so semilinear mapping is something which is  additive and satisfies this action. So alpha of A X equals fee of A times alpha of X. So  it's like linear mapping except that we have to twist the scalar by twisting morphism.  So in that case, we have also well-defined evaluation morphism consisting in mapping an  O-way polynomial F of X to F of alpha. Yeah.  Sorry.  Here the evaluation morphism of G of K or just K.  The morphism of evaluation. And the morphism of K. So I don't have any question.  Okay.  So you can evaluate not only at fee, but at any alpha, which is semilinear mapping acting  on some K vector space. And the proposition is that it's a classification proposition.  It's rather straightforward to prove this proposition. But the semilinear mapping of  K from K to K are just a multiplier of five. A times five for some A and K. And so using  this, you see what you can do. Instead of evaluating only at fee, we can just evaluate  at other alpha of this form and get this way of more general gabelian codes. So it's what  we call the linearized gabelian codes, which were defined by Martin Espinaz recently.  And also in a different spirit, with different notation by Delphine Boucher. But I will use  the presentation of Martin Espinaz, which is more suitable for what I wanted to say here,  today. So I consider as before K and M two integers. And A1, AM, elements of nonzero  elements of K. And I assume that the norm of the AIs, the norm of K over A are pairwise  distinct. So now the right condition. And I'm considering also V1, V2, Vm sub F vector  spaces of K. So I'll play the role of the V as a previous slide. And then the linearized  read salomon code, that's the city to this data, which is denoted by LRS, linearized  read salomon of K, A1, V1, A2, V2, AM, Vm. It's the easy major of this mapping. So you  take some array polynomial and we evaluate it at A1 phi, A2 phi, AM phi. So it's something  which makes the gabelian construction and the read salomon construction. So now the natural  points for evaluating array polynomials are these sublinear mappings. And I consider many  of these sublinear mappings, A1, A1 phi, A1 phi, A2 phi, A3. And I evaluate the array  polynomials at these points. Okay. And theorem is almost the same as before, is that the  length of this code is, so it's a computation is a bit, the computation is, so we have to  compute the dimension over K of the codomain of this map, which is a dimension of V1 plus  a dimension of V2 plus a dimension of Vm. Sorry. Yeah. Yes, because A1 phi is a mapping  from K to K. So the length is this. Yeah. So we can imagine that we are considering  more general sublinear morphism and yeah, but I'm just sticking to this. So the length  is n, the dimension is K, and again the minimal distance is n minus K plus 1. So say the minimal  distance for the aming distance, but actually it's a minimal distance for refined distances  before, which is called the same rank distance. So you have elements u1, um, ui lies is a  morphism from Vi to K, and now you samed the rank of all this ui. Okay. So it's the notion  of linearized read-salomon code, and I told you before that the gabin reading code were  used for network coding and linearized read-salomon codes are also application in the concrete  life and used for what we call the multi-shot linearized, multi-shot network coding. So  I don't want to say much about this because I don't know that much actually, but it's  not only a beautiful mathematical construction, it's also, it also has applications. Yeah.  So now I would like to, maybe I can give this in the blackboard for a moment, and I would  like to give you, to reinterpret this construction in a more general metric setting, spirit.  So for this I need to give, to explain more things on orypoenomials. So let's first, let  me first introduce this notation. R is the dimension of K over half. So it's the order  of phi. And I consider this ring, which is the ring of Laurent orypoenomial, I don't  know, so it's a orypoenomial in X and X minus one. And because phi is objective, it's not  a problem because we have this commutation relation, which is X minus one times A is  phi minus one of A times X minus one. So this ring has two important subrings. The first  one is F to the X to the R and X to the minus R, Laurent polynomials in F in the variable  X to the R. And it's actually the center of A. So it's quite, we'll play a quite important  role, you can check this, it's not, it's not difficult. And in between, maybe I will write  this again on the blackboard. So here you have A, which is K of X. Here you have the  center and in between you have C, which is K of X to the R and to the minus R. So this  is the center of A and C is the maximal sub-comutative sub-algebra. Okay. And what you can do is  you can extend scalar to C from Z to C. And in that case on the, here, this way. So you  have the tensor product. And since this C over Z is a Galois covering because it just  obtained by extending the scalar from K to F, which is a Galois extension, then C times  C, sorry, C tensor C over Z is just split as a product of copies of C, of R copies  of C, C to the R. And here you have C tensor A and it turns out that C tensor A are also  a very interesting interpretation. We have this isomorphism, C tensor A isomorphic to  the algebra of C linear transformation of A. So the mapping is following. Okay. So C  tensor A maps to the map, C acts by multiplication on the left and A acts by multiplication on  the right. And the important point is that A is a C module, of course, but which is free  of rank R. The basis, for instance, is 1 X, X square, X to the cube and X to the R minus  1. And so this is just a matrix algebra over C of dimension with R rows and R columns.  So A is some or a polynomial, but when we extend scalar from Z to C, then we get a matrix  algebra. And if you know algebraic geometry, Mike knows that this is a definition, almost  a definition of Azumaya algebra. So Azumaya algebra is an algebra, we become isomorphic  to a matrix algebra after an et al extension. And here's an extension C over Z is et al.  Because it just, it comes from, as I said before, it's a Galois covering, it comes from  the K over, it's deduced by scalar extension from the extension K over F, which is Galois.  So we have an Azumaya algebra and when you have an Azumaya algebra, you have a lot of  interesting facts. So for instance, we have what we call the radius trace coming from  the trace. So when you have a matrix algebra, you have the trace determinant and so on. And  this descend to A. So you have a radius trace, a radius trace map, which comes from A to  Z, I don't know, which is given by this explicit formula. It's not so important if you can  see the formula, just to say that it's explicit. And in the same way, you have the radius norm  and the radius norm corresponding to the determinant. And also you have an explicit formula only  for polynomial of degree one, otherwise much more complicated, but anyway, I won't show  them. And now I just rephrase this. So it's a commutative variable I had before. And now  I can just make a picture instead of diagrams. So I draw the spectrum of the center, spectrum  of the, just a straight line. It's A1 with 1.3 moved. I have a covering of spec Z by  spec C, which is a straight line, but defined over K, not over F. I have another copy of  spec C here, and now I have the tensor product. So it's a fiber product and is a Dijon student  of R copies of spec C, so R equals 2 in my drawing. And so above one point, I have two  points. And I can consider the vector space, the K vector space generated by these two  points, which are drawn here. And when I let the point vary on the axis, on the axis, I  can just make the construction vary on the, so the vector space move on the top. And what  I get is a vector bundle. And this isomorphism, saying that C tensor by A over Z is equal to  some matrix algebra, just says that C tensor A is the under morphism of this vector bundle.  And in fact, we can just, you can just let phi act on the under morphism of this vector  bundle, and so A can be recovered under morphism of this vector model stable by phi. So I don't  know how to draw spec A because A is non-commutative and I'm not really, I don't really know how  to draw the spectrum of a non-commutative algebra, so I just don't do it, but I'm doing the  under morphism of the vector bundle.  Can you say again what the vector bundle, what the fibers, it's a vector bundle over  C?  It's a vector bundle over C whose fiber is just, so it's a trivial vector bundle over  C of dimension R.  And if I have some point Z on the, on spec Z, I can try to figure out what is the fiber  of A over Z, which is by definition this quotient. And by the general theory of azimaya algebra,  you know that it's a simple central algebra over Z. The point is that this vector bundle  does not descend to C.  And so it's a simple central algebra, which is in general not under morphism of algebra,  just a simple central algebra, but for instance, if you are working over a finite field, all  simple central algebra are matrix algebra.  And in fact, we have characterization. This simple central algebra is split if and only  if Z is a norm in this extension.  In that case, yeah, so in that case, it means that if it splits, it's under morphism of some  vector space over Z. And so I'm just drawing this vector space here, but you have to be  very careful.  This is the vector bundle that doesn't descend on Z, so just the fiber here.  And so in that case, this isomorphism, I mean the splitting of this isomorphism is given  by this map, this map evaluation of A phi I've defined before.  So this evaluation of mapping f of X to f of A phi, realize this splitting. And so it's  a way to see that the fiber of A over Z acting on this vector space.  And so what is linearized with Salomon code, you just have some element of A, and each  time you have a point on spec Z, you can evaluate it.  You have this evaluation map, this evaluation map gives you an an anamorphism of K, and  K can be seen as something laying above Z.  So what you have to maybe to retain is that the gabidrin code actually is  something laying over, leaving over a point, the point one, we have a vector bundle of  dimension, vector space of dimension one over a point, and the linearized gabidrin code  is the same situation, but we just evaluate that not only at one, but at many other points.  Okay so it's a drawing.  And you can wonder if we can do this in a more general situation, instead of having  a covering of P1 by P1 or A1 with one point removed by A1 with one point removed, can  try to do this in a more geometric situation, we have a curve on the bottom and another  covering which is a cyclic Galois covering.  But it's actually a working progress, I won't say so much about this, but it's something  which would be linearized geometric codes or something like this.  So it's a very, very early stage, so I just started this work with Amore Drouin, we will  be a PhD student of mine next year.  So in the five, ten minutes I have, I will just want to speak about duality.  Maybe I won't finish what I want to say about this, but I think it's important.  So the starting point is this theorem.  If you have a read salomon code, which was defined, I remember by this evaluation formula  in the very classical setting, then we can define the dual of this code.  So by definition the dual is the orthogonal for the usual standard product of K to the  N, on K to the N, natural pairing.  And the dual of this code is the image of, can be described as the image of an anothomapping,  given in terms of residues.  This is quite interesting because the dual of a code in general is defined, is an intersection  of hyperplane, but it is described as an image of something else.  The proof of this is not difficult, it is a residue formula and an argument dimension.  But there is an interesting corollary that using this description you can prove that  the dual of read salomon codes actually also meet the single ton bound.  So not only the read salomon codes are MDS, but the dual of them are also MDS.  And what I want to do is try to generalize this result to a linearized read salomon code.  So if I have the dual of a linearized read salomon code, which is this code I defined  before, I want to have a description of the image of the dual.  So there are some difficulties.  So of course we are now looking for the image of something coming from, defined in terms  of orypoenomials.  The problem is that we don't know what is the residue for orypoenomials, we have to  define this.  And the second point is that now we no longer have something with the image in K to the  end, but in this andomorphism of K to the M, and so we have to define a pairing on this  space.  So there are two points to do, two things to do.  So I'm starting with a residue.  For this I need to figure out what is a good orypoenomials.  So the expansion is nothing but trying to evaluate orypoenomials on some thickening.  And so I say orypoenomials, andomorphism of some vector models.  The point is that on the left hand side, of course, this is a trigger vector model, you  can just consider the image of a thickening of a point, and then you get a thickening  of the vector models, no problem.  But on the right side, it's not so clear what you get.  And here comes a theorem saying that if you have a close point in spexies, that is an  orypoenomial in Z, then there is an isomorphism between this projective limit corresponding  to the function of the thickening and a ring of power series ring over the special fiber.  And okay, maps A and 2H.  And from this we can define a residue of orypoenomials.  So first of all I need to define the fraction field of A, because if I want a residue I  want a fraction.  So it's defined as A tensor of fractions, quotient.  So okay, I skip this.  And now I have the residue map, so I'm considering N a reducible polynomial in Z.  If I have a orypoenomial I can just map it to it, consider this Taylor expansion.  And I can use the previous theorem to see the Taylor expansion as a real, I mean as  a concrete series in H. And I can just consider the fraction field on the left, then it's  since N maps to H, it maps to this projective limit where N is inverted.  And on the right hand side it maps to a low run series with coefficient in A over N A.  And now I can consider the coefficient in H minus 1 as for the usual residues.  And I call this composite a residue at N.  And I can define the residue at A as the residue at this polynomial which is the point Z over  which A is defined and I evaluate at A is this.  And this way I get an endomorphism of K.  Okay, yeah, just a minute.  So I just want one slide.  And now I can define the pairing of the endomorphism of A for this I need to proceed by steps.  Okay, so I first define a pairing of K which is a usual pairing given by the trace.  Sorry.  Then I can define a pairing of the endomorphism of K by taking this usual formula, the trace  of the I joint of U composed with V.  And then I have a pairing on the mapping which is the sum of the trace.  Okay.  And this can actually be put back on our polynomials.  So the pairing is defined by this formula.  You see that the trace and the I joint and the turn out that the trace and the I joint  have some analogs on our polynomials.  So for the trace we have seen that we have a radius trace here and it's easy to prove  that the radius says it makes this I and I commute so it's when you evaluate.  So after evaluation the radius trace becomes a trace.  And it already himself define a pairing on our polynomials which is defined by this formula.  So we define a junction on our polynomial defined by this formula.  Sum of A I X I is the I joint of this is the sum of X minus Y A I.  And after evaluation this becomes a natural pairing.  This adjuncts becomes the I joint I defined before on the anamorphism of K.  And using this you see that the pairing of evaluation of G and evaluation of F at this  point is actually the radius trace of G star F evaluated at some point.  And you can yes, yes, yes, yes, finished.  And you have the same thing for the residue.  The pairing of the residue and evaluation is the residue of the radius trace of something.  And using this you can finally prove this theorem which is that the dual of the linearized  with Salomon code is the image of this, of a code like this, where you have the residue  at A I inverse of something like this.  And the corollary of this is that the dual of linearized with Salomon code means a singleton  bond as well as for with Salomon codes.  And I'm done.  Thank you.
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