Algebraic Coding Theory e-Summer School - ACT21
University of Zurich, Switzerland
June 7-11, 2021

--- In memory of Prof. Michele Elia ---

May 31, 2021 Program of ACT21 has been uploaded.
April 13, 2021 Registrations for ACT21 are open.
March 11, 2021 The 2021 ACT summer school will take place online between June 7 and June 11, 2021. The details about the program and registration will be announced soon.
March 11, 2020 The 2020 ACT summer school has been postponed by one year due to COVID-19 epidemic.

The landscape of algebraic coding theory has undergone major changes in the last years. In particular, the theory of error-correcting codes has inspired many mathematicians who were interested in applying techniques from algebra, algebraic geometry and discrete mathematics in order to progress on questions in information processing. Indeed, coding theory lies at the intersection of several disciplines in pure and applied mathematics such as algebra, number theory, probability theory, statistics, combinatorics, complexity theory, and statistical physics, which all have helped in the past to increase our knowledge in communication theory.

This summer school aims to focus the attention on different algebraic aspects of the theory of error-correcting codes. The summer school is structured in four mini-courses, each of which is intended to deepen a specific aspect of coding theory.

Group codes over fields are ideals in the group algebra KG, where K is a finite field and G is a finite group. Introduced by Berman and MacWilliams in the late sixties as a generalization of cyclic codes, they are still the subject of intense research. This short course is intended to be an introduction to their theory, presenting their main properties in relation to classical codes. The last part of the lecture will give an overview of current research perspectives and open problems. A good reference (among the rare ones) for these codes is Chapter 16 of the recent Concise Encyclopedia of Coding Theory by Huffman, Kim, and Solé.

This lecture is an introduction to the theory of rank-metric codes (linear spaces of matrices endowed with the rank metric). I will define these objects and describe their mathematical structure via a series of fundamental parameters, such as the minimum distance, weight distribution, and covering radius. I will then illustrate how these quantities interact with each other and how upper and lower bounds for them can be derived.

Galois Geometry is the study of the geometry of subspaces of a vector space over a finite field. In the projective geometry PG(n-1,q) we define points to be the 1-dimensional subspaces of an n-dimensional vector space over the field with q elements, lines to be the 2-dimensional subspaces, etc. Arrangements and intersection properties of sets of points, lines, hyperplanes etc have been studied for many years for a variety of reasons.
From the generator matrix of a linear code, in either the hamming or rank metric, we can use the columns to define geometric objects in an appropriate projective geometry, and intersection properties of these objects with hyperplanes are then related to the weights of codewords.
The correspondence between codes in the Hamming metric and (multi)sets of points in a projective geometry is well known and has been studied for many years; in particular the correspondence between MDS codes and arcs. The correspondence between (vector) codes in the rank metric and linear sets is more recent. In this lecture we will introduce these two correspondences, and illustrate how Galois Geometry and Coding Theory feed off each other, with results and techniques from one informing the other.

Codes over ring alphabets gained interest since the 1990s, when it was observed that several extremely good nonlinear binary codes may be viewed as linear codes over the ring ℤ4. In the following years, fundamental questions on code equivalence and the MacWilliams identity have led to a fruitful interplay between ring theory and coding theory. On a more practical side, novel algebraic decoding algorithms have been developed that are able to cope with the existence of zero divisors. More recently, codes over rings also received attention due to new applications in network coding scenarios.
In the lectures, I plan to provide an introduction to ring theory with a view towards coding applications, and I will also touch on some recent developments in ring-linear network coding and rank-metric codes.

In addition to the mini-courses, open problem sessions and contributed talk sessions are planned during the week, according to the detailed program schedule. The event is dedicated to junior researchers, doctorate students and Postdocs, as well as other established researchers that are interested in the topic.

### IMPORTANT DATES

April 2021 Registration Opens
June 7-11, 2021 Summer School

### PROGRAM OF THE SUMMER SCHOOL

Monday Tuesday Wednesday Thursday Friday
14:45 - 15:00 Opening
15:00 - 15:50 Martino Borello Jens Zumbrägel ACT Graduate
Minisimposium
Alberto Ravagnani John Sheekey
15:50 - 16:00 Coffee Break Coffee Break
16:00 - 16:50 Martino Borello Jens Zumbrägel Alberto Ravagnani John Sheekey
16:50 - 17:00 Coffee Break Coffee Break
17:00 - 17:50 Martino Borello Jens Zumbrägel Alberto Ravagnani John Sheekey
17:50 - 18:30 Get together on Wonder Get together on Wonder

The time indicated in the schedule is referred to Central European Summer Time (CEST).

15:00 - 15:25 Rank metric code invariants: a $$q$$-polymatroid approach Benjamin Jany, University of Kentucky Bilinear Complexity of 3-tensors Linked to Coding Theory Giuseppe Cotardo, University College Dublin Break On the list decodability of rank-metric codes Paolo Santonastaso, University of Campania An orbital construction of optimum distance Flag codes Miguel Ángel Navarro-Pérez, University of Alicante Break Analysis of Low-Density Parity-Check Codes over Finite Integer Rings for the Lee Channel Jessica Bariffi, German Aerospace Center (DLR) and University of Zurich Codes with good distance properties from a density perspectiveAnina Gruica, Technical University of Eindhoven Get together on Wonder

### Martino Borello

University Paris 8, France

CODES IN GROUP ALGEBRAS

### Alberto Ravagnani

Eindhoven University of Technology, the Netherlands

RANK METRIC CODES AND THEIR FUNDAMENTAL PROPERTIES

### John Sheekey

University College Dublin, Ireland

GALOIS GEOMETRY AND CODES

### Jens Zumbrägel

University of Passau, Germany

RING LINEAR CODING THEORY

### REGISTRATION

Registrations are closed.

### TECHNICAL INFO

The summer school will take place virtually using Zoom and Wonder.

You can use the following link to get help on using Zoom and Wonder platform:

Further information will be provided to the registered participants via email.

### COMMITTEES

Scientific Committee

Organizing Committee