Joseph Briggs: Extremal Collections of k-Uniform Vectors
Channel: Grad Student
Date: 2018-02-28

How many vectors can you have with ten 1's, and the rest 0's, whose span have dimension at most some large r? As well as being a fun problem in its own right, this question arises naturally in the context of both coding theory and matroids. In this talk, we will see some instances where all weight-10 vectors of length exactly r are the best we can do. Perhaps, that was the first example of such a collection of vectors you may think of-it appears to be remarkably robust across different (even infinite) fields, and even when swapping ``0'' with ``1''...but, once we go beyond $\mathbb{F}_2$, there are more letters to play with than just ``0'' and ``1'', so the number of variants on this question explodes! We can answer them all for $\mathbb{F}_3$, but not $\mathbb{F}_4$.