q10 — prescribed automorphisms

A construction family for $\ell_2(r,2)$ that does not start from a known covering. Instead of perturbing the certified 50-set, fix a $\sigma\in GL(r,2)$ of odd prime order $p$ and look only at $\sigma$ -invariant sets. Such a set is $k$ full orbits of size $p$ together with $m$ of the $2^c-1$ nonzero fixed vectors, so $n=pk+m$ — usually a very short list of possibilities, and the whole search collapses from $\binom{2^r-1}{n}$ to $\binom{\#\text{orbits}}{k} \binom{2^c-1}{m}$ .

Files

file	what it does
`setup.py`	the original $r=10$ , order-7, 1-dimensional-fixed-space instance: orbits, partner pairs, centraliser classes
`sigma_setup.py`	general $\sigma$ builder — emits a column table plus its centraliser classes on orbits and on partner pairs
`fixed_classes.py`	centraliser classes of $m$ -subsets of the fixed vectors
`orbit_search.c`	first search, $r=10$ order 7 hardcoded
`orbit_search_g.c`	general $r$ , fixed space of dimension $0$ or $1$ ; also carries the randomised orbit-level search
`orbit_search_f.c`	general $r$ and general fixed space
`layer_lemma.py`	per-layer counting bound
`prime_orders.py`	which prime orders are possible at all, and what each would cost
`run_q10_checks.sh`	replay: rebuilds every artifact, runs the controls, reruns every search

The reduction

Coverage is tracked orbit-wise. With $\Delta$ for sums,

source	covers
orbit $i\in S$	orbit $i$
sums inside orbit $i$	$\mathrm{orb}(r_i+\sigma^d r_i)$ , $d=1..p-1$
sums across $i\ne j$	$\mathrm{orb}(r_i+\sigma^d r_j)$ , $d=0..p-1$
orbit $i$ with fixed $g$	$\mathrm{orb}(r_i+g)$ — a single orbit
fixed $g\in S$	$g$
fixed $g$ with fixed $h$	$g+h$

Any of those sums can land on a fixed vector, which gets its own bit. Radius $\le2$ is exactly "every bit set", which is what the C searches decide.

Two counting bounds cut the case list before any search. $\sigma$ has odd order, so $V=M\oplus T$ canonically with $T$ the fixed space and the projection onto $T$ is $\pi_T(v)=\sum_{i<p}\sigma^i(v)$ . Call $\pi_T(v)$ the layer. Every sum above has a forced layer, which gives

touched layers (prime_orders.py): the layers a solution can reach lie in $\{0\}\cup\{t_i\}\cup\{t_i+t_j\}\cup\{t_i+g\}$ , so $1+k+\binom k2+km\ge2^c$ ;
layer fill (layer_lemma.py): a layer receiving no cross-orbit sum holds at most $k$ covered orbits (plus the internal sums when the layer is $0$ ), so if that is short of $L=\#\text{orbits}/2^c$ then every layer needs a cross pair and $\binom k2$ must cover them.

Controls

Encoding. The orbit masks reproduce the flat syndrome sweep hole for hole on hundreds of random selections, for every $\sigma$ used.
Positive. The same code finds $\sigma$ -invariant coverings at $n=63$ ( $r=10$ ) and $n=92,100,111$ ( $r=11$ ), each re-verified $1024/1024$ or $2048/2048$ by the independent flat sweep. The family is not vacuous.
Pruning. Rerunning $r=10$ , $n=49$ with --noprune visits all $\binom{144}{5}=481{,}008{,}528$ subsets per class and returns the same verdict.
Two implementations. orbit_search_g.c and orbit_search_f.c share no search code and agree on every case both can run.