## Presentation

Coding theory is a remarkable example of how various areas of mathematics can be used to solve problems in the storage and transfer of information. In this school, participants will be introduced to various techniques used in recent years to solve problems in coding theory. The aim is to motivate participants to do research in coding theory.

## List of courses

Course 1               :               Linear Block Codes, Dr Khumbo Kumwenda, Mzuzu University

Course 2               :               Finite Fields and Boolean Functions, Dr Augustine Musukwa, Mzuzu University

Course 3               :               Goppa Codes, Kondwani Magamba, Malawi University of Science and Technology

Course 4               :               Cryptography, Dr Mwawi Nyirenda, University of Malawi

Course 5               :               Codes and Designs, Dr Nephtale Mumba, Mzuzu University

Course 6               :               Codes, Graphs and Groups, Prof Eric Mwambene, University of the Western Cape

Course 7               :               Elliptic Curve Cryptography, Dr Ezekiel Kachisa, GIZ Malawi

Course 8               :               Linear Block Codes, Rank Metric Codes and their invariants, Prof Elisa Gorla, University of Neuchatel, Switzerland

Course 9              :           LCD Codes, Prof Bernardo Rodrigues, University of Pretoria, South Africa

Course 10            :           Self-Dual Codes, Prof Bernardo Rodrigues, University of Pretoria, South Africa

Course 11            :               Code-Based Cryptography, Prof Edoardo Persichetti, Florida Atlantic University, USA

## Abstracts

1.       Fundamentals of Coding Theory

Khumbo Kumwenda (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.

2.       Finite Fields and Boolean Functions

Augustine Musukwa

3.       Irreducible Goppa Codes

Kondwani Magamba (Malawi University of Science and Technology)

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\$.

4.       Cryptography

Mwayi Nyirenda (University of Malawi)

Cryptography is the science of hiding data from unwanted entities. There are two basic steps: encryption and decryption. There are some cryptographic algorithms that are building blocks, which are used to perform secure communication between authorized entities. This course takes you through the mathematical aspects behind these cryptographic algorithms while gaining knowledge of how they work. It covers the following: cryptosystem and its services (confidentiality, integrity, authentication, nonrepudiation, access control), some classical cryptosystem, some modern cryptosystem. We will also consider cryptanalysis of cryptographic systems and discuss recent trends in applications of cryptography. In this technology abreast era, cryptography has proven to be crucial in providing security to information. These will include but not limited to email security protocols, cybersecurity, cryptocurrency.

5.       Codes from Graphs

Nephtale Mumba (Mzuzu University, Malawi)

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.

6.       Codes, Graphs and Groups

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

The course focusses on combinatorial and algebraic structures that are useful in the study of codes from graphs.

7.       Elliptic Curve Cryptography

Dr Ezekiel Kachisa (GIZ Malawi)

8.       Linear Block Codes, Rank Metric Codes and their invariants

Prof Elisa Gorla, University of Neuchatel, Switzerland

9.       LCD Codes

Prof Bernardo Rodrigues (University of Pretoria, South Africa)