## Abstracts

**1.
****Coding in Distributed Systems**

** **

**Speaker: **Emina
Soljanin, Rutgers University

Many have
heard of Morse, bar, and QR codes, ISBN, and blockchains. These and other codes
play essential roles in many scientific disciplines and virtually all
telecommunication systems. We use codes to ensure reliability, security, and
privacy in data transmission and storage. In theory, we use codes to, e.g.,
study computational complexity, design screening experiments, and provide a
bridge between statistical mechanics and information theory. Codes can even
help understand the (quantum) spacetime fabric of reality. Some may use codes
for entertainment, e.g., to solve balance puzzles such as the penny weighing
problem. We can design social (hat color) guessing-game strategies that
significantly increase the odds of winning.

In this short course, the students will learn the fundamentals of coding theory
and practice and a selected number of more advanced topics. We will explore
coding in distributed systems. This topic is important because large volumes of
data collected for knowledge extraction must be reliably, efficiently, and
securely stored. We often distribute data over multiple nodes. Retrieval of
large data files from storage has to be fast (and often anonymous and private).
This course focuses on data storage and access and its relevant mathematical
disciplines, including algebraic coding and queueing theory.

**2.
****Rank
Metric Codes**

Speaker: Alberto
Ravagnani (Eindhoven University of Technology, Netherlands)

This course
is an introduction to the theory of rank-metric codes and their fundamental
parameters. I will explain how rank-metric codes can be used to correct errors
in network communications and introduce a series of mathematical tools to study
their structure and analyze their performance in applied contexts. The
prerequisite for this course are linear algebra and finite fields (the basics).

**3.
****Evaluation
Codes in Code-Based Cryptography**

Speaker: Gretchen
Mathews (Virginia Tech, USA)

In 1978,
Robert McEliece introduced a public key cryptosystem based on
the difficult problem of decoding a random linear code. Due to its large
key size, the McEliece cryptosystem has yet to see widespread use.
However, given that its security does not rely on factorization as the
commonly employed RSA and elliptic curve crypto systems do, it is now
being considered as a candidate for post-quantum cryptography. While
evaluation codes support many important applications of coding theory, they
must be modified for implementation in a code-based crypto system, such as that
based on McEliece. We will discuss why that is the case as well as possible
adaptations that are better suited for code-based cryptography.

**4.
****Fundamentals
of Coding Theory**

** **

**Speakers: **Khumbo Kumwenda &
Augustine Musukwa (Mzuzu University, Malawi)

This course will
introduce students to elementary results in the theory of error correcting
codes. It will look at construction, encoding and decoding of linear codes,
bounds in coding theory, cyclic codes and some special cyclic codes. As an
example, the course will focus on irreducible Goppa codes. In this regard, we
will study the parity check matrix and parameters of the codes and the parity
check matrix. An irreducible Goppa code of degree $r$ and length $q^n$ will be
defined in terms of a single field element and show that if one element that
defines a code can be transformed into another by a combination of an affine
map and Frobenius automorphism, then their corresponding codes are equivalent.
By categorizing these codes via the said maps and applying Cauchy Frobenius
Theorem, we produce an upper bound on the number of inequivalent irreducible
Goppa codes of degree $r$ and length $q^n$.

**5.
****LCD
Codes**

Bernardo Rodrigues
(University of Pretoria, South Africa)

This class builds on
work on Codes from Graphs. It focusses on linear codes with a complementary
dual (LCD codes). By considering codes from the row span of adjacency matrices
of graphs and associated reflexive and complementary graphs, it is shown that
we can find LCD codes with a particularly useful feature. As an application,
some classes of graphs are considered. These include the uniform subset graphs.
It is shown that if a p-ary code from a graph has this particular LCD feature,
the dimension of the code can be found from the multiplicities modulo p of the
eigenvalues of an adjacency matrix and, bounds on the minimum weight of the
code and the dual code follow from the valency of the graph

**6.
****Graphs
and Designs**

** **

Eric Mwambene
(University of the Western Cape, South Africa)

A lot of effort has been
directed towards the determination of properties of codes from the row span of
adjacency and incidence matrices of graphs and designs. In this lecturer, we
discuss various combinatorial and algebraic properties of graphs and designs
that make them useful for codes. These include coverings, graph products, graph
automorphisms, vertex and edge transitivity, connectivity, edge-cuts and
bipartitions. For the designs, we will mainly focus on properties of
neighbourhood and incidence designs of regular graphs.

**7.
****Codes
from Graphs**

** **

Nephtale Mumba (Mzuzu
University, Malawi)

In this class, we will
build on the course on graphs and designs. We will discuss the construction of
codes from graphs and associated designs. Looking at specific examples like
strongly-regular graphs, uniform subset graphs, some bipartite graphs, we will
determine properties of the codes. Our interest is in the length, dimension,
minimum distances and automorphism groups of the codes. We will also discuss permutation
decoding of the codes.

**8.
****Network
Coding**

Felice Manganiello (Clemon
University, USA)

The research in network
coding was initiated 20 years ago by a seminal paper by Ahlswede, Cai, and
Yeung. Since then researchers have studied many exciting network coding
problems, fundamental capacity limits and performance gains. The research in
this area has connections with other areas such as complexity theory, graph
theory, matroid theory, coding theory, and information theory. This course aims
at introducing networks and networks coding problems, and at teaching the
techniques that have been employed to solve them. The students will be provided
with enough mathematical knowledge to start researching in this applied area.

**9.
****Introduction
to Cryptography**

Mwayi Nyirenda-Kayuni
(University of Malawi)

The aim of this lecture
is to introduce basic principles of cryptography with a view to studying
code-based cryptography. We will study encryption, decryption, cryptosystems
and public key cryptography