Characterization of convexity for codes with more than four maximal codewords

Determine whether, for neural codes with more than four maximal codewords, the joint absence of Giusti–Itskov local obstructions and Ruys de Perez et al. wheels is sufficient and necessary to characterize convexity.

Background

The paper confirms Jeffs’ conjecture for many cases with up to four maximal codewords and shows additional results for minimal L24 codes. It raises a broader question of whether the same obstruction-based characterization extends to codes with more than four maximal codewords, noting NP-hardness of convexity detection and the utility of combinatorial tools like local obstructions and wheels.

A potential starting point is a collection of 96 five-maximal codes on six neurons that are minimal (hence free of local obstructions) but whose convexity status and wheel presence are unknown.