## Paulo Almeida

## Superregular matrices for MDP convolutional codes

In the transmission of digital information, the transmitted message can be corrupted by noise. Coding is a technique which consists of the introduction of redundancy in the message, in order to allow the detection and correction of the errors which occurred during the message transmission. Codes with good error correcting properties are the ones with large distance. The codes whose column distances increase as rapidly as possible for as long as possible are called maximal distance profile (MDP) codes. These codes are very appealing since they have the potential to have a maximal number of errors corrected per time interval in sequential decoding.

In this talk we focus on convolutional codes. These codes are used in a variety of systems, such as wireless and satellite communications. However, the problem of how to construct MDP codes is not completely solved and very little is known about the minimal field size required for doing so. It turns out that this issue is based on the construction of a particular type of superregular matrices.

Superregular matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure. We will present a new construction of superregular matrices over a sufficiently large field that can be used to obtain MDP codes. We also provide a bound on the required field size needed for such matrices to be superregular.

In this talk we focus on convolutional codes. These codes are used in a variety of systems, such as wireless and satellite communications. However, the problem of how to construct MDP codes is not completely solved and very little is known about the minimal field size required for doing so. It turns out that this issue is based on the construction of a particular type of superregular matrices.

Superregular matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure. We will present a new construction of superregular matrices over a sufficiently large field that can be used to obtain MDP codes. We also provide a bound on the required field size needed for such matrices to be superregular.

António Breda

## Pseudo-regular maps and hypermaps

Regular maps and hypermaps have been the main focus of research for many years. Regularity however, is not absolute and can sometimes be taken more broadly. In this talk I will speak about pseudo-regularity introduced for maps by Wilson in the eighties, and the classification of pseudo-regular hypermaps of small genus that Rui Duarte, Domenico Catalano and I embarked upon a few months ago.

Elisa Fernandes

## Polytopes of hight rank for the symmetric and the alternating groups

An abstract regular polytope is a string C-group, that is, a group generated by a finite ordered set of involutions, satisfying the intersection condition and such that nonconsecutive involutions commute. The rank of an abstract regular polytope is the number of involutions of the generating set. We use the term r-polytope when referring to an abstract regular polytope of rank r.

In a recent work with Dimitri Leemans we proved that there is a unique (n−1)-polytope and also a unique (n−2)-polytope for the symmetric group Sym(n). In this proof we use classical theorems on permutation groups. In this talk I will give the idea of this proof.

For the alternating group Alt(n) we constructed families of polytopes of rank equal to (n−1)/2 when n is odd and (n−2)/2 when n is even. We conjectured that, in both cases, this is the maximal rank of a polytope with automorphism group Alt(n). I intend to explain how these high rank families are constructed and to give the standing point on the proof of the maximal rank conjecture.

In a recent work with Dimitri Leemans we proved that there is a unique (n−1)-polytope and also a unique (n−2)-polytope for the symmetric group Sym(n). In this proof we use classical theorems on permutation groups. In this talk I will give the idea of this proof.

For the alternating group Alt(n) we constructed families of polytopes of rank equal to (n−1)/2 when n is odd and (n−2)/2 when n is even. We conjectured that, in both cases, this is the maximal rank of a polytope with automorphism group Alt(n). I intend to explain how these high rank families are constructed and to give the standing point on the proof of the maximal rank conjecture.

Patrícia Ribeiro

## Towards the standard spherical tiling

*isometric folding*is a map that sends piecewise geodesic segments into piecewise geodesic segments of the same length. Given two isometric foldings f and g of a surface M, f is said to be deformable into g if there exists a continuous map H : [0, 1] ×M → M, such that, for each t ∈ [0, 1], H

_{t}is an isometric folding.

It is known, since 1989, that any non-trivial isometric folding of the Euclidean plane is deformable into the standard planar folding defined by f(x, y) = (x, |y|). However, the correspondent situation on the sphere remains an open question.

Related to spherical isometric foldings are

*spherical*f

*-tilings*, that is, edge to edge decompositions of the sphere by geodesic polygons, such that all vertices are of even valency and both sums of alternate angles, around any vertex, is π. The relation between these two set of objects comes from the fact that the set of singularities of any non trivial isometric folding be a spherical f-tiling.

As expected, the problem of isometric folding deformations gives rise to a similar problem of spherical f-tiling deformations. More precisely, is any spherical f-tiling deformable into the standard tiling (f-tiling whose underline graph is a great circle)?

Unfortunately, the deformation of isometric foldings does not induce a deformation of its associated f-tilings (the set of singularities) and so the Hausdorff metric is not enough. In order to overcome this problem, a new metric arose (2005)

*T*(S

^{2}) by giving to each face of a spherical f-tiling a convenient orientation.

Here, we provide a way to deforme into the standard f-tiling, three distinct classes of dihedral f-tilings of the sphere whose prototiles are two non congruent spherical triangles.

This is a joint work with Ana Breda.