How many ways are there to write down n nonnegative integers, all of which being strictly smaller than q, such that their sum is k? In other words, what is the coefficient of x^k in (1 + x + ... + x^(q-1))^n?

In this thesis, we are going to dive into the integer partitioning problem with a certain number of partitions and an upper bound on partition size.

The goal of this thesis is to take an already existing bound for the value mentioned above, which holds for a specific k value - and extend it to general k. 

The upper bound can then be used for proving better upper bounds in coding theory for certain metrics other than the Hamming metric.

Voraussetzungen

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. 

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.