arXiv:2610.10973v1 Announce Type: cross Abstract: Let $k>t\ge 2$ be fixed integers, and let $G$ be a permutation group on a set of $v$ points. A $G$-invariant packing is free if every block has $|G|$ distinct images under $G$. We give two conditions on $G$ under which, as $v\to\infty$, there is a free $G$-invariant $t$-$(v,k,1)$ packing with $(1-o(1))\binom{v}{t}/\binom{k}{t}$ blocks. This is a $G$-invariant form of the Erd\H{o}s--Hanani theorem on asymptotically optimal packings. For semiregular groups $G$, packings with these properties exist if and only if all but $o(v^t)$ of the $t$-subsets of points have trivial setwise stabiliser. This property is satisfied by every semiregular group for $t\ge 3$, and by every cyclic semiregular group for $t=2$. As a consequence, the Johnson bound is asymptotically attained at all lengths by optical orthogonal codes of constant weight $w$ and constant correlation $\lambda$ with $w\ge\lambda+2$, by their multidimensional and signature pattern versions, and by cyclic and quasi-cyclic constant-weight codes. The same method gives a form of the theorem for group divisible packings that are invariant under a cyclic group. This determines the asymptotic maximum size of two-dimensional optical orthogonal codes with at most one pulse per wavelength.