Matrix-rank considerations for odd primes via analogous p-designs

Determine whether the matrix-rank arguments used in this paper—based on incidence matrices of 2-(2q−1, q−1, q/2−1) designs to prove full-rank properties of inclusion matrices—extend to odd primes p by constructing and analyzing analogous p-designs that yield full-rank inclusion matrices for the corresponding settings.

Background

The paper establishes simplicity results for Steinberg and groupoid C*-algebras associated with Z2-multispinal groups by demonstrating that certain inclusion matrices have full rank. This is achieved by identifying an infinite family of 2-(2q−1, q−1, q/2−1) designs (with q a positive even integer) and leveraging their incidence matrices to obtain the required rank properties.

The authors note that the distinction in simplicity depending on the characteristic of the underlying field appears in related contexts and that their approach bridges groupoid algebra methods with combinatorial design theory. However, they explicitly state that they do not know whether similar matrix-rank considerations can be made for other primes, as they have not identified suitable p-designs for odd primes p. This raises a concrete question about constructing and validating analogous design structures for odd primes that would support comparable rank arguments.