Nontrivial kernels of triple Khatri–Rao products

Determine whether the kernel of the triple Khatri–Rao product h_i^{\ast 3} is nontrivial or trivial in the random setting for the product-expanding local classical codes used to construct good non-Abelian sheaf qLDPC codes.

Background

The gauged sheaf-code construction requires a nonzero cycle \xi in a tensor-product coefficient complex. For local parity-check matrices h_i, the existence of such a cycle requires \ker h_i^{\ast 3}\neq 0, where h_i^{\ast 3} is the column-wise triple Khatri–Rao product. However, the local codes must also have sufficiently large distance, which generally requires large matrix dimensions and may eliminate the kernel. The paper notes that probabilistic constructions of the required two-way product-expanding codes do not currently resolve whether this kernel remains nontrivial in the random case.