Optimal distributions for Hamming-code blowups

Determine which distributions on \(\mathbb Z_2^k\setminus\{(0,\dots,0)\}\), equivalently which part ratios in blowups of the \((2^k-1,2^k-k-1)\)-Hamming code, maximize the probability that \(d\) independently drawn vectors contain a basis.

Background

The paper obtains sharp statistics constants for several small pairs (d,s)(d,s) using blowups of Hamming codes. This motivates considering blowups of the general (2k−1,2k−k−1)(2^k-1,2^k-k-1)-Hamming code for configurations of size 2d−k2^{d-k}. The unresolved issue is the optimization of the distribution of the nonzero vectors, or equivalently the relative sizes of the parts in the blowup, for the event that independently sampled vectors span the full vector space.

References

However, it is unclear in general which distributions on $ Z_2k\setminus{(0,\dots,0)}$ (which correspond to part ratios in blowups) maximise the probability that $d$ independently drawn vectors contain a basis.

— Some exact values of the inducibility and statistics constants for hypercubes  (2503.03408 - Bodnár et al., 5 Mar 2025) in Section 5, Concluding Remarks