# Powerful sets: a generalisation of binary matroids

## TL;DR

This paper introduces powerful sets as a generalization of binary matroids, exploring their properties, constructions, and the fact that most are nonlinear, expanding the understanding of combinatorial structures related to binary codes.

## Contribution

It defines powerful sets, investigates their combinatorial properties, and shows that almost all such sets are nonlinear, extending the theory beyond traditional binary matroids.

## Key findings

- Every powerful set is determined by its minimal nonzero members.
- The number of powerful sets is doubly exponential.
- Almost all powerful sets are nonlinear.

## Abstract

A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary vectors as characteristic vectors of subsets of $E$, we say that a set $S\subseteq2^E$ of subsets of $E$ is a \textit{powerful set} if the set of characteristic vectors of sets in $S$ is a powerful code. Powerful sets (codes) include cocircuit spaces of binary matroids (equivalently, linear codes over $\mathbb{F}_2$), but much more besides. Our motivation is that, to each powerful set, there is an associated nonnegative-integer-valued rank function (by a construction of Farr), although it does not in general satisfy all the matroid rank axioms. In this paper we investigate the combinatorial properties of powerful sets. We prove fundamental results on special elements (loops, coloops, frames, near-frames, and stars), their associated types of single-element extensions, various ways of combining powerful sets to get new ones, and constructions of nonlinear powerful sets. We show that every powerful set is determined by its clutter of minimal nonzero members. Finally, we show that the number of powerful sets is doubly exponential, and hence that almost all powerful sets are nonlinear.