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

Abstract: For a finite graph $H$, a connectivity graph-code is a family $\mathcal C\subseteq 2<sup>{E(H)}$ such that $A\triangle B$ is a connected spanning subgraph of $H$ whenever $A$ and $B$ are distinct members of $\mathcal C$. Let $m(H)$ denote the maximum size of such a family, and let $f(d)$ be the largest integer $q$ for which $m(H)=q$ for infinitely many pairwise nonisomorphic $d$-regular graphs $H$. Restricting codewords to the edges incident with a vertex gives $f(d)\leq 2<sup>d$. Alon proved equality for all sufficiently large $d$ and asked whether it holds for every $d\geq 4$. We answer this question affirmatively. More precisely, for every $d\geq 4$ we construct infinitely many finite simple $d$-regular bipartite graphs carrying a linear connectivity graph-code of dimension $d$. The construction begins with a vector-labelled copy of $K_{d,d}$. For $d\geq 7$, the required labelling follows from a probabilistic count over an irreducible conjugacy class in $\mathrm{GL}_d(2)$; explicit matrices, verified by a short exact exhaustive program, cover $d=4,5,6$. Cyclic voltage lifts then produce the required infinite families.