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tags: Assignments
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# Assignment 5

:::info
Due Friday October 28
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This assignment is an investigation, a chance to do research, with the solution not known in advance. Hence the student should submit a report on what they did, supporting GAP code, and any findings that they came across. The report is limited to 5 pages, and it does not need to fulfill 5 pages (it can be short).

## Description of the mini-research project

Recently, [Bamberg, Giudici, Lansdown, and Royle](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms.12619) used the theory of association schemes to show that a particular group was a synchronising group of diagonal type; the first such example ever found. To keep a long story short, they showed that every nontrivial graph $\Gamma$ of the conjugacy class scheme of $PSL(2,13)$ has $\omega\cdot \alpha<|PSL(2,13)|=1092$ where $\omega$ is the clique number of $\Gamma$, and $\alpha$ is the coclique number of $\Gamma$. They essentially used Delsarte's LP-bound, together with some ad-hoc methods. The goal of this research project is to see whether a spherical code bound can show that some graphs $\Gamma$ have this property. If the student so chooses, they can change the group involved to see if there are other groups which have this property, and extend the problem in whichever direction they choose.

## Objectives

Using the spherical code technique may or may not work, your task is to find out. The report should document everything you do to get there, so that:
1. it is easy to verify your conclusion;
2. we can see the processes you went through in conducting your research.

You could write your report in the style of a lab report (e.g., Aims, Methods, Conclusion, Results).

## Resources

- [Delsarte, Goethals, Siedel](https://link.springer.com/article/10.1007/BF03187604)
- [Baller et al.](https://arxiv.org/abs/1606.06620)