Previous speakers

Title: The minimum distance of the code of intersecting lines in PG(3,q)

Abstract: Let L denote the set of all lines in PG(3,q), with q=p^h, p prime. For every subset S of L, define the characteristic function of S as

X_S: l in L ----> 1 if l is in S; 0 otherwise.

For each line l, we denote by X_l the characteristic function of the set of lines that intersect l. Consider the linear code 

C(q) = < X_l| l in L >

of F_p^{L}. We call this linear code C(q) the code of intersecting lines in PG(3,q). We  determine the minimum weight of this code. For q even, this minimum weight is q^3+q^2+q+1, and the minimum weight codewords are the incidence vectors of the sets of totally isotropic lines of a symplectic polarity in PG(3,q). For q odd, the minimum weight is q^3+2q^2+q+1, and the minimum weight codewords are the non-zero scalar multiples of the vectors X_l, with l in L. We also prove that this type of incidence vectors  are equal to the codewords of second weight in the code of intersecting lines in PG(3,q), q even.

Title: Foundations of hyperbolic geometry - Slides 

Abstract: 

The independent discovery by Lobachevsky and Bolyai of hyperbolic geometry in the 1830's was followed by slow acceptance of the subject from the 1860's on, with the publications of relevant parts of the correspondence of Gauss. A new phase was entered from 1903, when David Hilbert, in his work introducing the "calculus of ends", introduced an axiomatisation for hyperbolic plane geometry by adding a hyperbolic parallel axiom to the axioms for plane absolute geometry. In 1938, Karl Menger (of the famous Vienna Circle) made the important discovery that in hyperbolic geometry the concepts of betweenness and equidistance can be defined in terms of point-line incidence. Since an axiom system obtained by replacing all occurrences of betweenness and equidistance with their definitions in terms of incidence would look highly unnatural, Menger and his students looked for a more natural axiom system. In particular, Helen Skala showed in 1992 that there is a set of axioms whose models are the classical hyperbolic planes over Euclidean fields, and her axioms were the first that contained only first order statements. This talk will be on joint work with Tim Penttila (Emeritus, University of Adelaide) where we endeavour to simplify Skala's axioms and retain a characterisation of the classical hyperbolic planes.

Title: Back to the Future - Slides - Video

Abstract: 

This talk tells the history of coding theory through the lens of Reed Muller codes. In the beginning, there were no computers, and coding theory was the mathematics of sphere packing. This was a golden time for algebraic coding, with the discovery of Reed Muller and Reed Solomon codes. As everyday computers became more powerful coding theory changed character and focused on iterative algorithms. Today with quantum computers on the horizon, Reed Muller codes are back in fashion.

Title: The asymptotics of r(4,t) - Slides 

Abstract: 

For integers s,t > 1, the Ramsey numbers r(s,t) denote the minimum N such that every N-vertex graph contains either a clique of order s or an independent set of order t. I will give an overview of recent work, joint with Jacques Verstraete, which shows

r(4,t)=Ω(t^3/log^4(t)) as t -> ∞.

This determines r(4,t) up to a factor of order log^2(t), and solves a conjecture of Erdős. Moreover, I will discuss some 

subsequent work with David Conlon, Dhruv Mubayi and Jacques Verstraete showing the need for good constructions, possibly coming from finite geometry.

Title: Grassmannians of codes - Slides 

Abstract: Abstract 

In this talk I will consider the point line-geometry P_t(n,k) having as points all the [n,k]-linear codes having minimum dual Hamming weight at least t+1 and where two points X and Y are collinear whenever the intersection X andY is a [n,k-1]-linear code having minimum dual Hamming weight at least t+1.

Let L_t(n,k) be the collinearity graph  of P_t(n,k). Then L_t(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph D_t(n,k) of the linear codes having minimum dual Hamming weight at least t+1 introduced in [2].

I will investigate the structure of L_t(n,k) focusing on its relation with well-studied configurations of points of a projective space such as the saturated sets. In particular, I will characterize the set of isolated vertices of L_t(n,k) and for t=1 and t=2,  necessary and sufficient conditions for L_t(n,k) to be connected will be provided. Finally, these results will be applied to the geometry P_t(n,k) in order to study its projective embeddability by means of the Plücker map.

This is a joint work with Luca Giuzzi.

[1]  I. Cardinali and L. Giuzzi, Grassmannians of codes, submitted.

[2]  M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263.

Title: Evasive sets, covering by subspaces, and point-hyperplane incidences - Slides 

Abstract:

Given positive integers k<= d and a finite field F, a subset S of F^d is (k,c)-subspace evasive if every k-dimensional affine subspace contains at most c elements of S.

By a simple averaging argument, the maximum size of a (k,c)-subspace evasive set is at most c |F|^d-k.

In this talk we discuss the construction of evasive sets, matching this bound.

The existence of optimal evasive sets has several interesting consequences in combinatorial geometry.

Using it we determine the minimum number of k-dimensional linear hyperplanes needed to cover the grid [n]^d.

This extends the work by Balko, Cibulka, and Valtr, and settles a problem proposed by Brass, Moser, and Pach.

Furthermore, we improve the best known lower bound on the maximum number of incidences between points and hyperplanes in dimension d assuming their incidence graph avoids the complete bipartite graph K_{t,t} for some large constant t=t(d).

Joint work with Istvan Tomon.

Title: Properties of the Direct Sum of q-Matroids -- Slides 

Abstract:

After a brief introduction of q-matroids and their relevance for rank-metric codes we will survey some of the main results in the still young theory of q-matroids. They comprise an extensive list of cryptomorphisms. While these are non-trivial results, they all form quite natural q-analogues of the corresponding cryptomorphisms for (classical) matroids. We will then turn to the direct sum of q-matroids, which was introduced in 2021 by Ceria/Jurrius. It turns out that the definition as well as the properties of the direct sum are significantly different from those for matroids. After discussing the construction of the direct sum, we will report on properties where the theory diverges the most from that of matroids. Thereafter, we will turn to a result where the theory, surprisingly, meets that of matroids. Indeed, the direct sum behaves very naturally with respect to cyclic flats. This allows us to show that every q-matroid can be decomposed into irreducible ones and to characterize irreducibility.

This is joint work with Benjamin Jany.

Title: Consequences of a resultant-like theorem in Galois geometries -- Slides -- Video 

Abstract:

See abstract or click in the link below!

Title: Sidon spaces -- Slides 

Abstract:

Motivated by a problem related to difference sets, Hou, Leu and Xiang introduced  in 2002 a linear version of the classical theorem of Kneser in additive combinatorics, where sets are replaced by subspaces and cardinalities  by dimensions. A nice feature of the linear version is that, via  Galois extensions, it provides an alternate proof of the original version. This openned a trend to prove extensions of theorems in additive combinatorics to their linear analogues. The talk will focus on one of these extensions, the Vosper theorem, which gives rise to the notion of Sidon spaces. This notion turned out to find interesting applications in coding theory.

This is joint work with Christine Bachoc and Gilles Zémor, with a nice simplification by Chiara Castello.

Title: On Cameron-Liebler sets of k-spaces in finite projective spaces (Part I)

Abstract:

This is part 1 (of 2) of a double talk together with Jonathan Mannaert. Cameron-Liebler line classes in a finite 3-dimensional space PG(3,q) originate from the study by Cameron and Liebler in 1982 of groups of collineations with equally many orbits on the points and the lines of PG(3,q). These objects have some interesting equivalent characterizations, and are examples of Boolean functions of degree one. One of the main properties of this set is that these line classes admit a parameter x, which can be used to classify or exclude examples. In this talk, we focus on these objects from a geometric perspective, and report on several existence and non-existence results, including a recent so-called modular equality for the parameter of Cameron-Liebler line classes in finite n-dimensional projective spaces found in [2] for n odd. This modular equality is a natural generalization of the modular equality found in [3].

[1] A. Blokhuis, M. De Boeck, and J. D'haeseleer.

Cameron-Liebler sets of k-spaces in PG(n,q).

Des. Codes Cryptogr., 87(8):1839--1856, 2019.

[2] J. De Beule and J. Mannaert.

A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension.

Submitted.

[3] A. L. Gavrilyuk and K. Metsch.

A modular equality for Cameron-Liebler line classes.

J. Combin. Theory Ser. A, 127:224--242, 2014.

Title: Algebraic curves with many automorphisms -- Slides

Abstract:

The Hurwitz upper bound on the size of the K-automorphism group Aut(C) of an algebraic curve C of genus g greater than 1 defined over a field K of zero characteristic is 84(g-1). 

In positive characteristic p, algebraic curves can have many more automorphisms than expected from the Hurwitz bound.  

There even exist algebraic curves of arbitrarily large genus g with more than 16g^4 automorphisms. Besides the genus, an important invariant for curves in positive characteristic is the p-rank of the curve, which is the integer c such that the Jacobian of C has p^c p-torsion points. It turns out that the most anomalous examples of algebraic curves with a very large automorphism group invariably have zero p-ranks.

Several results on the interaction between the automorphism group, the genus and the p-rank of a curve can be found in the literature. In this talk we survey some reults on the following issues that have been obtained in the last decade:

(i) Upper bounds on the size of Aut(C) depending on g and the structure of Aut(C).

(ii) The possibilities for Aut(C) when the p-rank is 0.

(iii) Upper bounds on the size of d-subgroups of Aut(C). 

Some applications to maximal curves over finite fields are also discussed.

Title: The geometry of extremal Cayley graphs

Abstract:

The geometric aspect of extremal Cayley graphs is highlighted, providing a different proof of known results and giving a new perspective on how to tackle such problems.

Some new results about extremal  pseudrandom triangle free graphs are also presented.

Title: Small strong blocking sets and their coding theoretical counterparts

Abstract:

Strong blocking sets are sets of points in the projective space such that the intersection with each hyperplane spans the hyperplane. They have been defined first in Davydov, Giulietti, Marcugini, Pambianco, 2011, in relation to covering codes, and reintroduced later as generator sets in Fancsali, Sziklai, 2014 and as cutting blocking sets in Bonini, Borello, 2021, in relation with minimal codes. In Alfarano, Borello, Neri, 2019 and independently in Tang, Qiu, Liao, Zhou, 2019, it has been shown that strong blocking sets are the geometric counterparts of such codes. From their definition, it is clear that adding a point to a strong blocking set maintains the property of being strong, so that strong blocking sets of small cardinality are the most interesting ones. In the coding theoretical language, this is equivalent to have a short minimal code. A natural question is then how small a strong blocking set in a projective space of a given dimension can be.

 In the talk, we will illustrate these connections, together with some bounds on their parameters and with some constructions of small strong blocking sets. At the end, we will describe some perspectives and analogues in the rank metric.

Title: The dynamics of iterating functions over finite fields --- Slides 

Abstract:

When we iterate functions over finite structures, there is an underlying natural functional graph. For a function f over a finite field Fq, this graph has q nodes and a directed edge from vertex a to vertex b if and only if f(a)=b. It is well known, combinatorially, that functional graphs are sets of connected components, components are directed cycles of nodes, and each of these nodes is the root of a directed tree from leaves to its root.

The study of iterations of functions over a finite field and their corresponding functional graphs is a growing area of research, in part due to their applications in cryptography and integer factorization methods like Pollard rho algorithm. 

Some functions over finite fields when iterated present strong symmetry properties. These symmetries allow mathematical proofs of some dynamical properties such as period and preperiod of a generic element, (average) "rho length" (number of iterations until a cycle is formed), number of connected components, cycle lengths, and permutational properties (including the cycle decomposition).

We survey the main problems addressed in this area so far. We exemplify by describing the functional graph of Chebyshev polynomials over a finite field. We use the structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials over finite  fields. We conclude providing a list of open problems. 

Based on joint works with Rodrigo Martins (UTFPR, Brazil), Claudio Qureshi (UdelaR, Uruguay) and Lucas Reis (UFMG, Brazil).

Title: Vectorial bent functions and beyond --- Abstract --- Slides 

Abstract: 

A function F:F2^n -> F2^m is called vectorial bent if  (x+a)+F(x)=b for all a different from 0 and all b has exactly 2^{n-m} solutions.

It is well known that n=2k must be even and that m<= k.

In my talk, I will address some problems about the classification of vectorial bent functions, in particular:

• Classification of (6,3)-vectorial bent functions [1].

• Number of quadratic (n,2)-vectorial bent functions [3].

Due to the bound m<= k, one may ask which functions are close to vectorial bent functions if m>k. In  [2] we determined the maximum number of bent functions that may occur as component functions of F:F2^{2k} -> F2^{2k}. It turns out that this maximum is 2^k and the non-bent functions form a vector space (bent complement). This has been later generalized to functions F:F2^{2k}->F2^m [4].

I will briefly report about recent progress on such MNBC functions (joint work with Bapić,  Pasalic and Polujan). 

References

[1] A. A. Polujan and A. Pott, On design-theoretic aspects of Boolean and vectorial bent function, IEEE Trans. Inform. Theory, 67 (2021), pp. 1027–1037.

[2] A. Pott, E. Pasalic, A. Muratović-Ribić, and S. Bajrić, On the maximum number of bent components of vectorial functions, IEEE Trans. Inform. Theory, 64 (2018), pp. 403–411.

[3] A. Pott, K.-U. Schmidt, and Y. Zhou, Pairs of quadratic forms over finite fields, Electron. J. Combin., 23 (2016), pp. Paper 2.8, 13.

[4] L. Zheng, J. Peng, H. Kan, Y. Li, and J. Luo, On constructions and properties of (n, m)-functions with maximal number of bent components, Des. Codes Cryptogr., 88 (2020), pp. 2171–2186.

Title: Eigenvalues of oppositeness graphs and Erdős-Ko-Rado for flags --- Slides 

Abstract: 

Over the last few years, Erdős-Ko-Rado theorems have been found in many different geometrical contexts including for example sets of subspaces in projective or polar spaces. A recurring theme throughout these theorems is that one can find sharp upper bounds by applying the Delsarte-Hoffman coclique bound to a matrix belonging to the relevant association scheme. In the aforementioned cases, the association schemes turn out to be commutative, greatly simplifying the matter. However, when we do not consider subspaces of a certain dimension but more general flags, we lose this property. In this talk, we will explain how to overcome this problem, using a result originally due to Brouwer. This result, which has seemingly been flying under the radar so far, allows us to find eigenvalues of oppositeness graphs and derive sharp upper bounds for EKR-sets of certain flags in projective spaces and general flags in polar spaces and exceptional geometries. We will show how Chevalley groups, buildings, Iwahori-Hecke algebras and representation theory tie into this story and discuss their connections to the theory of non-commutative association schemes.

Title: Linear Fractional Transformations and Irreducible Polynomials over Finite Fields --- Slides 

Abstract: 

We will discuss polynomials over a finite field where linear fractional transformations permute the roots. For subgroups G of PGL(2,q) we will demonstrate some connections between the field of G-invariant rational functions and factorizations of certain polynomials into irreducible polynomials over Fq. Some unusual patterns in the factorizations are explained by this connection.

Title: Designs over finite fields by difference methods

Abstract: 

At the kind request of the organizers, I will try to give an outline of how difference methods allow to obtain some q-analogs of 2-designs. Of course, a particular attention will be given to the renowned 2-analog of a 2-(13,3,1) design found by Braun, Etzion, Östergård, Vardy and Wassermann.

Title: The tensor rank of semifields of order 81

Abstract: 

Tensor products of vector spaces are fundamental objects in mathematics. The tensor product of two vector spaces can be studied using matrices, and this case is well-understood; the rank can be calculated easily, and equivalence corresponds precisely with rank. However for higher order tensors, problems such as calculating the rank or determining equivalence becomes very difficult.

The case of the tensor product of three isomorphic vector spaces corresponds to algebras in which multiplication is not assumed to be associative. In this case, the tensor rank gives an important measure of the complexity of the multiplication in the corresponding algebra. For the case of a finite semifield (i.e. a not-necessarily associative division algebras), lower bounds can be obtained using results from linear codes, while for field extensions upper bounds can be obtained via polynomial interpolation and algebraic geometry.

In this talk we will survey these problems and present new results where we determine the tensor rank of all finite semifields of order 81. In particular we show that some semifields of order 81 have lower tensor rank than the field of order 81, the first known example of such a phenomenon.

This is joint work with Michel Lavrauw.

Title: How many lines of the Fano plane do we need to color a cubic graph? --- Abstract --- Slides 

Abstract: 

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is related to a long standing conjecture in graph theory attributed to Berge and Fulkerson.  

It turns out that such a problem can be nicely described in term of colorings of the edge-set of the graph by using as colors the points of suitable configurations in PG(2,2) and PG(3,2) (see [1]). 

More precisely, given a set T of lines in the finite projective space PG(n,2), a T-coloring of a cubic graph G is a coloring of the edges of G by points of PG(n,2) such that the three colors occurring at any vertex form a line in T.

In the first part of the talk we present the main problem in its original formulation and we show the connection with T-colorings.

Then, we present some recent results (see [2]) on a minimum possible counterexample for the Berge-Fulkerson Conjecture.

[1] E. Máčajová, M. Škoviera, Fano colourings of cubic graphs and the Fulkerson Conjecture, Theor. Comput. Sci. 349 (2005) 112-- 120.

[2] E. Máčajová, G. Mazzuoccolo, Reduction of the Berge-Fulkerson conjecture to cyclically 5-edge-connected snarks, Proc. Amer. Math. Soc. 148 (2020), 4643-4652.

Title: On certain pairs of primitive elements on finite fields --- Slides 

Abstract: 

In this talk we would like to present some results on the existence of pairs of elements in a finite field,  where the first element is either primitive or primitive and normal over a subfield, and the second element is primitive and a rational function of the first one. 

This is based on joint works with J.P Guardieiro, V. Neumann and G. Tizziotti.

Title: On the hardness of the code equivalence problem in rank metric --- Slides 

Abstract: 

In this talk, we discuss the code equivalence problem in rank metric. For F_{q^m}-linear codes, which is the most commonly studied case of rank metric codes, we prove that the problem can be solved in polynomial case with an algorithm which is "worst case". On the other hand, the problem can be stated for general matrix spaces. In this situation, we are able to prove that this problem is at least as hard as the monomial equivalence for codes endowed with the Hamming metric.

This is a common work with Thomas Debris Alazard and Philippe Gaborit.

Title: Combinatorially defined point sets of finite Desarguesian planes --- Slides 

Abstract: 

Let S be a point set of PG(2,q). A line m is called a k-secant of S, if it meets S in exactly k points. Many of the famous objects of PG(2,q) have the property that each of their points is incident with the same number of k-secants, for every integer k. For example arcs, unitals, subplanes, maximal arcs and Korchmáros-Mazzocca arcs are such objects. In my talk I will present some characterization results of point sets with this property.

I will also introduce the following problem of a similar flavour. Let M be a point set of AG(2,q), q=p^n, p prime, and call a direction (d) uniform, if more than half of the lines with slope d meet M in the same number of points modulo p. We will call this number the typical intersection number at (d). The rest of the affine lines with slope d will be called renitent. Note that we allow different uniform directions to have different typical intersection numbers. I will show structural properties of the renitent lines, in particular I will show that they are contained in some low degree algebraic curves of the dual plane.

The talk is based on joint works with Simeon Ball, Péter Sziklai and Zsuzsa Weiner. 

Title: Maximal curves over finite fields --- Abstract ---  Slides 

Abstract:

Algebraic curves over a finite field $\mathbb{F}_q$ and their function fields have been a source of great fascination for number theorists and geometers alike, ever since the seminal work of Hasse and Weil in the 1930s and 1940s. 

Many important and fruitful ideas have arisen out of this area, where number theory and algebraic geometry meet. For a long time, the study of algebraic curves and their function fields was the province of pure mathematicians. But then, in a series of three papers in the period 1977-1982, Goppa found important applications of algebraic curves over finite fields to coding theory. 

The key point of Goppa's construction is that the code parameters are essentially expressed in terms of arithmetic and geometric features of the curve, such as the number $N_q$ of $\mathbb{F}_q$-rational points and the genus $g$.

Goppa codes with good parameters are constructed from curves with large $N_q$ with respect to their genus $g$. 

Given a smooth projective, algebraic curve of genus $g$ over $\mathbb{F}_q$, an upper bound for $N_q$ is a corollary to the celebrated Hasse-Weil Theorem,

$$N_q \leq q+ 1 + 2g\sqrt{q}.$$

Curves attaining this bound are called $\mathbb{F}_q$-maximal. The Hermitian curve $\mathcal{H}$, that is, the plane projective curve with equation 

$$X^{\sqrt{q}+1}+Y^{\sqrt{q}+1}+Z^{\sqrt{q}+1}= 0,$$

is a key example of an $\mathbb{F}_q$-maximal curve, as it is the unique curve, up to isomorphism, attaining the maximum possible genus $\sqrt{q}(\sqrt{q}-1)/2$ of an $\mathbb{F}_q$-maximal curve. Other important examples of maximal curves are the Suzuki and the Ree curves.

It is a result commonly attributed to Serre that any curve which is $\mathbb{F}_q$-covered by an $\mathbb{F}_q$-maximal curve is still $\mathbb{F}_q$-maximal. In particular, quotient curves of $\mathbb{F}_q$-maximal curves are $\mathbb{F}_q$-maximal. Many examples of $\mathbb{F}_q$-maximal curves have been constructed as quotient curves $\mathcal{X}/G$ of the Hermitian/Ree/Suzuki curve $\mathcal{X}$ under the action of subgroups $G$ of the full automorphism group of $\mathcal{X}$.

It is a challenging problem to construct maximal curves that cannot be obtained in this way for some $G$. 

In this presentation, we will describe our main contributions to the theory of maximal curves over finite fields.

In particular, the following topics will be discussed:

1.  how can we decide whether a given $\mathbb{F}_q$-maximal curve is a quotient of the Hermitian curve?

2. further examples of maximal curves that are not quotient of the Hermitian curve;

3.  determination of the possible genera of $\mathbb{F}_q$-maximal curves, especially quotients of $\mathcal{H}$;

4. Weierstrass semigroups on maximal curves.

Joint work with: Daniele Bartoli, Peter Beelen, Massimo Giulietti, Leonardo Landi, Vincenzo Pallozzi Lavorante, Luciane Quoos, Fernando Torres, Giovanni Zini.

Title: On linear systems of conics over finite fields --- Abstract --- Slides 

Abstract: 

A form on an n-dimensional projective space ${\mathbb{P}}^n$ is a homogeneous polynomial in $n+1$ variables. The forms of degree $d$ on ${\mathbb{P}}^n$ comprise a vector space $W$ of dimension ${n+d}\choose{d}$. Subspaces of the projective space ${\mathbb{P}} W$ are called linear systems of hypersurfaces of degree $d$.

The problem of classifying linear systems consists of determining the orbits of such subspaces under the induced action of the projectivity group of ${\mathbb{P}}^n$ on ${\mathbb{P}}W$. In this talk we will focus on linear systems of quadratic forms on ${\mathbb{P}}^2$ over finite fields. We will give an overview of what is known and explain some of the recent results. This is based on joint work with T. Popiel and J. Sheekey.

Title: Network Coding, Rank-Metric Codes, and Rook Theory

Abstract: 

In this talk, I will first propose an introduction to network coding and its methods. In particular, I will explain how codes with the rank metric naturally arise as a solution to the problem of error amplification in communication networks (no prerequisite in information theory is needed for this part). 

The second part of the talk concentrates instead on the mathematical structure of codes with the rank metric and its connection with topics in contemporary combinatorics. More precisely, I will present a link between rank-metric codes and q-rook polynomials, showing how this connection plays a role in the theory of MacWilliams identities for the rank metric.

Title: Curves over finite fields and polynomial problems

Abstract: 

Algebraic curves over finite fields are not only interesting objects from a theoretical point of view, but they also have deep connections with different areas of mathematics and combinatorics.

In fact, they are important tools when dealing with, for instance, permutation polynomials, APN functions, planar functions, exceptional polynomials, scattered polynomials.

In this talk I will present some applications of algebraic curves to the above mentioned objects.

Title: Defining Reed--Muller codes in the rank metric: the Alon--Füredi theorem for endomorphisms

Abstract:

Codes in the rank metric have gained a huge interest in the last years, due to their applications to network coding and cryptography. The most celebrated family of rank-metric codes is given by Gabidulin codes. It is well-known that they can be seen as analogues of Reed-Solomon codes in classical coding theory, which are codes constructed from spaces of univariate polynomials. The generalization of Reed-Solomon codes to multivariate polynomials lead to the family of Reed-Muller codes. In the last years, several researchers tried to adapt a Reed-Muller-type construction in the rank metric setting, unfortunately without success.  Hence, finding such a construction has been an open problem for several years. 

We observed that the main obstruction for constructing Reed-Muller codes in the rank metric was the impossibility to have abelian Galois extensions which are not cyclic, when dealing with finite fields. Motivated by this intuition, in this talk we switch to general infinite fields, and present the theory of rank-metric codes over arbitrary Galois extension. In the abelian case, we derive the analogues of the celebrated Alon-Füredi theorem and of the Schwartz-Zippel lemma for endomorphisms. These results provide nontrivial lower bounds on the rank of a linear endomorphism and are of independent interest. Moreover, they allow to show that we can construct rank-metric codes that share the same parameters with classical Reed-Muller codes. Central tool for this approach is the Dickson matrix associated to an endomorphism, which we carefully investigate.