Previous speakers

Title: Sequences of integers and generalized weights - Slides 

Abstract: 

Generalized weights are a well-studied family of invariants, originally defined for linear block codes in 1977. Their importance stems from the fact that they characterize the code robustness against wiretapping. In this talk we will investigate which sequences of integers can be the generalized weights of various types of error-correcting codes: linear block codes, rank-metric codes, and the more general case of sum-rank metric codes. We will answer this question also in the case of greedy weights and relative generalized weights.

Title: Bounded Degree- Low Rank Parity Check Codes

Abstract: 

Low Rank Parity Check Codes (LRPC codes) are the rank-metric analogue of low density parity check codes.

We introduced a new constrain on the support of the parity check matrix. In particular we require that the parity check matrix has its support in the Fq-linear space V_a,d = <1,a,a^2,...,a^d-1>.

It is easy to show that LRPC codes of density 2 (i.e. LRPC such that the support of their parity check matrix has dimension 2) correspond to BD-LRPC of bounded degree 2.

Thanks to the special structure of the subspace V_a,d, we proved that BD-LRPC codes with bounded degree d can uniquely correct errors of rank weight r when n − k ≥ r + u for certain u ≥ 1, in contrast to the condition n − k ≥ dr required for the standard LRPC codes.

The probability of failure of the algorithm we propose is exponential in q^{-u+1}.

As the code length n approaches infinity, when n/m → 0, it is shown that u can be chosen as certain constant, which indicates that the BD-LRPC codes with a code rate of R can be, with a high probability, uniquely decodable with the decoding radius ρ = r/n attaining the Singleton bound 1 − R.

Title: Tensor representation of semifields and commuting polarities

Abstract: 

We use the cyclic model for threefold tensors to investigate a geometric interpretation of finite semifields. In general dimension, this leads to a bound for a semifield invariant called BEL-rank. In the two dimensional case, this approach incorporates a new proof, purely geometrical, of a classical result by Dickson. Also, it naturally defines a pair of commuting polarities in PG(3, q^2). Using these polarities, we construct new quasi-polar spaces.

This is a joint work with John Sheekey.

Title: LRC codes and a construction from towers of function fields - Slides 

Abstract: 

In this work, we construct sequences of locally recoverable AG codes arising from function fields. In particular we study this construction for a tower of function fields and give bounds for the parameters of the obtained codes.  We will show an example from a tower over Fq2 for any odd q, defined by Garcia and Stichtenoth, and we will show that the bound is sharp for the first code in the sequence.

Title: On higgledy-piggledy sets and the André/Bruck-Bose representation - Slides 

Abstract: Abstract in pdf

In this talk, we focus on higgledy-piggledy sets of k-subspaces in PG(N,q), i.e. sets of projective subspaces that are "well-spread-out". More precisely, the set of intersection points of these k-subspaces with any (N-k)-subspace S of PG(N,q) spans S itself. In other words, the set of points in the union of these k-subspaces forms a strong blocking set w.r.t. (N-k)-subspaces. Naturally, one would like to find a higgledy-piggledy set consisting of a small number of k-subspaces.

Although these combinatorial sets of subspaces are sporadically mentioned in older works, only since 2014 researchers have started to investigate these sets as a main point of interest. In this talk, we aim to discuss the state of the art concerning this special type of subspace sets. Moreover, we want to present some recent results, some of which are joint work with Jozefien D'haeseleer and Geertrui Van de Voorde and concerns a higgledy-piggledy plane set in PG(5,q). The proof of its existence relies on the field reduction, linear sets and the André/Bruck-Bose representation of the projective plane PG(2,q^3).

Title: Weierstrass semigroups at the F_{q^2}-rational points of a maximal curve with the third largest genus

Abstract: Abstract in pdf

An F_{q^2}-maximal curve X of genus g is defined to be a projective, geometrically irreducible, non-singular algebraic curve defined over F_{q^2} such that the number of its F_{q^2}-rational points attains the Hasse-Weil upper bound.

F_{q^2}-maximal curves, especially those with large genus, are of particular interest in coding theory since they give rise to excellent AG codes.

It is well known that, for an F_{q^2}-maximal curve X, g(X) <= q(q - 1)/2 and that it reaches this upper bound if and only if X is F_{q^2}-isomorphic to the Hermitian curve. The first and the second largest genera of F_{q^2}-maximal curves are known, and they are realized by exactly one curve up to F_{q^2}-isomorphism, but the same is not clear for the third largest genus. Its value is known to be equal to g_3=\lfloor (q^2-q+4)/6 \rfloor, but it is still unclear whether this is realized by exactly one curve up to F_{q^2}-isomorphism. 

In this talk, I will present our results on the Weierstrass semigroups at the F_{q^2}-rational points of the curve X_3: x^{(q+1)/3} + x^{2(q+1)/3} + y^{q+1} = 0, with q \equiv 2 mod 3, which is a curve known to have genus equal to g_3. One of the surprising results is that there are roughly (q+1)/3 possible different semigroups, although not all of them may occur for a given q. Moreover, the curve X_3 has many non-F_{q^2}-rational Weierstrass points.

Joint work with Peter Beelen and Maria Montanucci.

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

Abstract: 

This is part 2 of a double talk together with Jan De Beule. Cameron-Liebler line classes in PG(n,q) are well studied objects due to several equivalent definitions and interesting properties, yet they appear to be scarce. These objects can be generalized naturally to Cameron-Liebler sets of k-spaces in PG(n,q), which inherit many properties. It is known that these sets of k-spaces are also examples of sets arising from Boolean degree 1 functions. Each Cameron-Liebler set of k-spaces has a parameter x. 

Conditions on this parameter yield non-existence results. In this talk we focus on general non-existence results for these sets of k-spaces, we do this by proving a lower bound on the parameter of non-trivial examples of Cameron-Liebler sets of k-spaces. The main techniques we apply arise from generalizaing techniques used in [2]. In his thesis, Drudge wanted to classify Cameron-Liebler line classes in PG(n,q) using their intersection with subspaces. In [1], these concepts where generalized to Cameron-Liebler sets of k-spaces and they also improve the known results obtained for Cameron-Liebler line classes in sufficiently large projective spaces.  

[1] J. De Beule, J. Mannaert and L. Storme.

Cameron-Liebler k-sets in subspaces and non-existence conditions.

Des. Codes Cryptogr., 90: 633–651, 2022.

[2] K. Drudge.

Extremal sets in projective and polar spaces.

PhD thesis, The University of West Ontario, London, Canada, 1998.

Title: Scattered linear sets in a finite projective line, translation planes and hyper-reguli of F_{q^t}^2

Abstract: 

In [2], G. Lunardon and O. Polverino show that the point set of a scattered F_q-linear set of rank t in PG(1,q^t), also called maximum scattered linear set (MSLS for short), is a derivable partial spread of the F_q-vector space F_{q^t}^2

(elsewhere such structures are also called hyper-reguli). Hence any MSLS gives rise to a non-Desarguesian translation plane. In the case dealt with in [2], the authors  obtain an André plane.

In this talk, a quasifield associated with any MSLS will be exhibited. Our main contribution is to prove that two translation planes associated with two MSLSs L_U and L_{U'} are isomorphic if and only if they are related to F_q-subspaces U and U' of F_{q^t}^2 belonging to the same orbit under the action of ΓL(2,q^t). As a consequence, any MSLS L_U gives rise to a set of pairwise nonisomorphic translation planes whose size is the ΓL-class of L_U, as defined in [1].

This is a joint work with V. Casarino and C. Zanella.

[1] B. Csajbók, G. Marino, O. Polverino:

Classes and equivalence of linear sets in PG(1,q^n),

J. Combin. Theory Ser. A 157 (2018), 402-426.

[2] G. Lunardon, O. Polverino:

Blocking sets and derivable partial spreads,

J. Algebraic Combin. 14 (2001), 49-56.

Title: Snarks and perfect matchings  --- Abstract --- Slides 

Abstract: 

Snarks, which for us represent bridgeless cubic graphs which are not 3-edge-colourable (Class II), are crucial when considering conjectures about bridgeless cubic graphs, and, if such statements are true for snarks, then they would be true for all bridgeless cubic graphs. One such conjecture which is known for its simple statement, but still indomitable after half a century, is the Berge-Fulkerson Conjecture which states that every bridgeless cubic graph G admits six perfect matchings such that every edge in G is contained in exactly two of these six perfect matchings. In this talk we discuss two other related and well-known conjectures about bridgeless cubic graphs, both consequences of the Berge-Fulkerson Conjecture which are still very much open: the Fan-Raspaud Conjecture (1994) and the S4-Conjecture (Mazzuoccolo, 2013).

Given the obstacles encountered when dealing with such problems, many have considered trying to bridge the gap between Class I and Class II bridgeless cubic graphs by looking at invariants that measure how far Class II bridgeless cubic graphs are from being Class I. This is done in an attempt to further refine the class of snarks, and thus, enlarging the set of cubic graphs for which such conjectures can be verified. In this spirit we consider a parameter which gives the least number of perfect matchings (not necessarily distinct) needed to be added to a bridgeless cubic graph such that the resulting multigraph is Class I. We show that the Petersen graph is, in some sense, the only obstruction for a bridgeless cubic graph to have a finite value for the parameter studied. We also relate this parameter to already well-studied concepts: the excessive index, and the length of a shortest cycle cover of a bridgeless cubic graph. 

The above is joint work with Edita Máčajová, Giuseppe Mazzuoccolo and Vahan Mkrtchyan.

Title:  Galois subcovers of the two Skabelund maximal curves --- Slides 

Abstract: 

In 2016 D. Skabelund constructed two maximal curves over finite fields as cyclic covers of the Suzuki and Ree curves. The two curves have been later investigated by M. Giulietti, M. Montanucci, L. Quoos and G. Zini, who determined the full automorphism group and computed the genera of many Galois subcovers of the two curves. This talk will give an overview of a recent work, in collaboration with P. Beelen and M. Montanucci, in which we completed the classification of all Galois subcovers of the two Skabelund maximal curves. The talk will focus on some of the techniques involved in the genus computation of such Galois subcovers, that lead to obtain new values in the spectrum of genera of maximal curves.

Title:  Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields --- Abstract --- Slides  

Abstract: 

Given an incidence structure (P, B), we say that a family F contained in B is intersecting if any two  elements of F share at least one point. There have been ample investigations into the size and structure of the largest intersecting families in a wide variety of incidence structures. We say that an incidence structure satisfies the strong EKR property if all intersecting families of maximum size consist of all the blocks through a fixed point.

In this talk I will discuss this problem in ovoidal circle geometries. They arise by taking a quadratic surface Q in PG(3,q) (which is a slight generalisation of a classical polar space) and taking the plane sections with every plane that intersects Q in an oval. I will discuss the proof that the strong EKR property holds in Möbius planes of even order greater than two, and in ovoidal Laguerre planes. As a corollary, the strong EKR property also holds for polynomials of bounded degree over a finite field.

The proof is an illustration of the beautiful marriage of Erdős-Ko-Rado problems and algebraic graph theory.

Title:  Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes --- Slides 

Abstract: 

A  pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q^n,q^n) and a Laguerre plane of order q^n (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q^-(5,q), non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q^-(5,q) is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and our work is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q^-(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. Joint work with John Bamberg and Alessandro Siciliano)

Title: The Sparseness of MRD Codes --- Slides 

Abstract: 

An open question in coding theory asks whether or not MRD codes with the rank metric are dense as the field size tends to infinity. For answering this question, I will discuss the problem of estimating the number of common complements of a family of subspaces over a finite field in terms of the cardinality of the family and its intersection structure. Upper and lower bounds for this number will be derived, along with their asymptotic versions as the field size tends to infinity.

By specializing these results to matrix spaces, one obtains upper and lower bounds for the number of MRD codes. In particular, I will show that MRD codes are sparse for almost all parameter sets as the field size grows. The new results in this talk are joint work with Alberto Ravagnani.

Title: Algebraic curves and (one of) their applications --- Slides 

Abstract: 

The foundation of the theory of algebraic curves over the complex field goes back to the Nineteenth century, and most of this theory holds true if C is replaced by any field of characteristic zero. However,  significant differences arise in positive characteristic. One of the main features of algebraic curves in positive characteristic concerns the fact that they may have much larger automorphism groups (compared to their genus) than in the zero characteristic case. 

A part of this seminar will be dedicated to the description of the relationship between automorphism groups and other birational invariants of an algebraic curve, and to the presentation of our main contributions.

Apart from their intrinsic theoretical interest, algebraic curves over finite fields have relevant applications to several areas of Mathematics.

In particular, in the last decades, methods of Algebraic Geometry have been prominent in Coding Theory with the so-called AG codes.

In fact, the essential ingredients for the computation of the parameters of AG codes are Riemann-Roch spaces and Weierstrass semigroups. We will give an overview of this topic and present some recent results.

Joint work with Massimo Giulietti, Gábor Korchmáros, Stefano Lia, Gábor Nagy.