Let $ {X}$ be the orthogonality graph, that is the graph with $ {\{ -1, 1 \}^n}$ as vertices with two vertices adjacent if they are orthogonal. So $ {x, y \in \{ -1, 1 \}^n}$ are adjacent if $ {x \cdot y = x_1y_1 + x_2y_2 + \ldots + x_ny_n = 0}$. There are […]
As Anurag Bishnoi likes to point out on his blog, an often overlooked source of wisdom is Willem Haemer’s PhD thesis from 1979. Many of Haemer’s proofs rely on simple properties of partitions of Hermitean matrices. My motivation for this post was a small exercise for myself. I wanted to prove the easy one of […]
At the moment I am very busy writing things like grant applications and research papers, so I lack the time for blog posts. But then I wasted part of my evening reading this article on FiveThiryEight about the Democratic Party primaries in the US. For each democratic primary contender, they provide the following data: How […]
I spent the last few days in vain using several spectral arguments to bound the size of certain intersection problems. For instance what is the largest set of vectors in $ {\{ 0, 1 \}^4}$ pairwise at Hamming distance at most $ {2}$ (a problem solved by Kleitman, recently investigated by Huang, Klurman and Pohoata). […]
Let us consider the $ {n}$-dimensional hypercube $ {\{ 0, 1 \}^n}$. The Hamming graph on $ {H_n}$ has the elements of $ {\{ 0, 1 \}^n}$ as vertices an two vertices are adjacent if their Hamming distance is one, so they differ in one coordinate. It is easy to see that the independence number […]
Last week I had a cold and could not do much thinking. So I spent my time making TikZ pictures for an upcoming talk of mine. This talk is on my recent work with Akihiro Munemasa on constructing cospectral strongly regular graphs. I think that the pictures are nice for a blog post, so here […]
This is the announced post on my recent paper with Yuval Filmus on Boolean degree $ {1}$ functions on association schemes. The post will focus on what motivates the problem from various points of view. Before I start, a small remark. I am using latex2wp for the first time in this post. Thanks to Luca […]
Permutation Groups, Analysis of Boolean Functions, Finite Geometry, Coding Theory and Algebraic Graph Theory Important mathematical concepts get reinvented many times. In my recent work with Yuval Filmus we explored objects that are called (in random ordering) Boolean degree 1 functions, Cameron-Liebler line classes, equitable partitions, completely regular strength 0 codes with covering radius 1, […]
This is a small report on a failed project — obtaining semidefinite programming bounds on constant dimension network codes. But let us start with some context … A network code consists of a set of subspaces in [latex]\mathbb{F}_q^n[/latex]. It is a code, so we want to maximize the distance between subspaces (or increase the code’s […]
