This thesis is concerned with developing code constructions for a new weight function (and associated metric) on (Z/qZ)^n called the unit-burst weight, suitable for measuring same-symbol burst errors. A unit burst is defined as a vector that has some consecutive positions of ones and is zero otherwise.

Any vector v in (Z/qZ)^n can be written as a (not necessarily unique) linear combination of these bursts. The burst weight is then the minimum number of bursts that need to be added or subtracted to produce v.

This metric has a connection to the A_n root lattice, a special lattice in Z^n+1 of vectors whose entries sum to zero. More precisely, the unit bursts relate to the shortest vectors of A_n called roots, and the burst weight corresponds to the smallest decomposition of a lattice point into roots.

We have already derived some basic properties and algorithms for this new metric and now would like to find some bounds and code constructions achieving those bounds. 

Rank-matric codes are codes that live in a vector space that is endowed with a different metric than the Hamming metric: in the rank-metric the distance between two codewords, represented as matrices over a smaller field, is defined as the rank of their difference.

The theory of rank-metric codes has interesting connections to many topics in discrete mathematics and promissing applications to code-based cryptography and network coding.

In this seminar work, the student will study properties and constructions of rank-metric codes and one or more applications. The goal is to understand and summarize parts of the following papers:

[1] H. Bartz, L. Holzbaur, H. Liu, S. Puchinger, J. Renner, A. Wachter-Zeh (2022). “Rank-Metric Codes and Their Applications”. arXiv: 2203.12384  https://arxiv.org/pdf/2203.12384.pdf 

[2] K. Marshall, (2016). “A study of cryptographic systems based on Rank metric codes”, Dissertation, University of Zurich  https://www.zora.uzh.ch/id/eprint/127105/1/Diss%20Kyle.pdf

[3] T. Randrianarisoa, (2018). “Rank metric codes, codes using linear complexity and applications to public key cryptosystems”, Dissertation, University of Zurich  https://www.zora.uzh.ch/id/eprint/153545/1/153545.pdf

[4] E. Gorla (2019). “Rank-metric codes”. arXiv: 1902.02650  https://arxiv.org/pdf/1902.02650.pdf

[5] E. Gorla and A. Ravagnani. (2018). “Codes endowed with the rank metric”. In: Network Coding and Subspace Designs. Springer. 3–23.  https://link.springer.com/content/pdf/10.1007/978-3-319-70293-3.pdf 

In this thesis, the student will study and summarize the most recent code-based digital signature schemes based on the syndrome decoding problem

In this thesis, the student will study the mathematics of linear codes and how they can be used to design cryptosystems.