Resources

Series: 11 of 12 articles available

A Series Learning through the MacWilliams Identity

1. Part 1 of 12An Introduction to Möbius Inversion through the MacWilliams Identity
2. Part 2 of 12An Introduction to Character Theory through the MacWilliams Identity
3. Part 3 of 12An Introduction to Tensor Products through the MacWilliams Identity
4. Part 4 of 12An Introduction to Krawtchouk Polynomials through the MacWilliams Identity
5. Part 5 of 12An Introduction to Association Schemes through the MacWilliams Identity
6. Part 6 of 12An Introduction to Pless Moments through the MacWilliams Identity
7. Part 7 of 12An Introduction to Matroids and Tutte Polynomials through the MacWilliams Identity
8. Part 8 of 12An Introduction to Group Actions and Cycle Indices through the MacWilliams Identity
9. Part 9 of 12An Introduction to Factor Graphs and Partition Functions through the MacWilliams Identity
10. Part 10 of 12An Introduction to Heat Kernels and Trace Formulas through the MacWilliams Identity
11. Part 11 of 12An Introduction to Lattices and Theta Functions through the MacWilliams Identity

Published: May 13, 2026

Generalised Vector Spaces and Finitary Matroids

This note rewrites the notion of a generalised vector space, which I learnt about from alg-d's video, in terms of matroid closure operators. We check that generalised vector spaces are essentially the same as finitary matroids, or pregeometries, and form a narrower class than modern infinite matroids in general.

Published: May 9, 2026

A Single Axiom for Matroid Closure Operators

This note states that a single equation for a map $\operatorname{cl} \colon 2^{E} \to 2^{E}$ on a finite set characterises the closure operator of a matroid.