Conference Agenda

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Session Overview
Session
MS180, part 1: Network coding and subspace designs
Time:
Thursday, 11/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F-113
53 seats, 70m^2

Presentations
10:00am - 12:00pm

Network coding and subspace designs

Chair(s): Daniele Bartoli (University of Perugia), Anna-Lena Horlemann-Trautmann (University of St. Gallen, Switzerland)

This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

More on exceptional scattered polynomials

Daniele Bartoli
University of Perugia

Let f be an F_q-linear function over F_(q^n). If the F_q-subspace U= { (x^(q^t), f(x)) : x in F_(q^n) } defines a maximum scattered linear set, then we call f a scattered polynomial of index t. We say a function f is an exceptional scattered polynomial of index t if the subspace U associated with f defines a maximum scattered linear set in PG(1, q^(mn)) for infinitely many m. There is a very interesting link between maximum scattered linear sets and the so called maximum rank distance (MRD for short) codes. In particular, a scattered polynomial over F_(q^n) defines an MRD code in (F_q)^(nxn) of minimum distance n-1. Exceptional scattered monic polynomials of index 0 (for q>5) and of index 1 have been already classified. In this work, we investigate the case t>1.

 

The size of linear sets on a finite projective line

Jan de Beule
University of Brussels

A linear set in a finite projective space, i.e. a finite dimensional projective space over a finite field, is a set of points whose defining vectors belong to an additive subgroup of the underlying vector space of the projective space.

Linear sets were introduced by G. Lunardon in the context of the construction of small blocking sets of finite Desarguesian projective planes. Meanwhile, linear sets have been used to construct and/or characterize many substructures of finite projective spaces. Recently, J. Sheekey described a correspondence between certain MRD codes and scattered linear sets on the finite projective line.

After giving briefly some properties of linear sets, and connections with other objects like blocking sets and MRD codes, we will report on joint work with Geertrui Van de Voorde in which we showed a lower bound on the number of points of a linear set on a finite projective line.

 

Rank Metric Codes and Subspace Codes in a Convolutional Setting

Joachim Rosenthal
University of Zurich

Subspace codes have been introduced by Koetter and Kschischang in order to tackle coding problems in the area of random linear network coding. Subspace codes are subsets of a fixed Grassmannian defined over a finite field. This class of codes is also closely related to the class of ``rank metric codes''.

In a first part we will show how rank metric codes induce in a natural way so called rank metric convolutional codes.

We will then report about some basic properties of rank metric convolutional codes.

In a second part we will show how rank metric convolutional codes can be lifted to subspace convolutional codes.

 

Partitions of Matrix Spaces and q-Rook Polynomials

Alberto Ravagnani
University College Dublin

I will describe the row-space and the pivot partition on the space of n x m matrices over GF(q). Both these partitions are Fourier-reflexive and yield invertible MacWilliams identities for matrix codes endowed with the row-space and the pivot enumerators, respectively. Moreover, they naturally give rise to notions of extremality. Codes that are extremal with respect to any of these notions satisfy strong rigidity properties, analogous to those of MRD codes.

The Krawtchouk coefficients of both the row-space and the pivot partition can be explicitly computed using combinatorial methods. For the pivot partition, the computation relies on the properties of the q-rook polynomials associated with Ferrers diagrams, introduced by Garsia/Remmel and Haglund in the 80's. I will describe this connection between codes and rook theory, and present a closed formula for the q-rook polynomial (of any degree) associated to an arbitrary Ferrers board.

The new results in this talk are joint work with Heide Gluesing-Luerssen.