Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$

What the math gives to ML

The paper constructs binary families of subgraphs whose pairwise XOR is always connected and spanning, attaining the bound of 2^d codewords on K_{d,d} for every d >= 4. The transferable asset is an algebraic generator for many sparse binary masks with a strong pairwise structural invariant, rather than the extremal graph theorem itself. This can be tested in dynamic sparse layers or MoE routing by ensuring that every mask transition preserves global communication between all units. The construction is compact: a d-bit index selects one of 2^d masks, while an irreducible binary operator generates the edge labels.