Papers

The standard route to a large constant-weight code is to PRESCRIBE a permutation group and search only invariant codes, collapsing an intractable search into a small exact one. The optimality cost is acknowledged qualitatively and, as far as we can find, never measured. We measure it. Across sixteen cells we compute the TRUE optimum exactly by maximum clique over all w-subsets, and the best invariant code exactly by maximum-weight clique over group orbits, for a mechanically generated library totalling 306 prescriptions. Three findings. First, prescription has no intrinsic ceiling at these parameters: in every one of the sixteen cells some group in the library attains the true optimum exactly, so the gap is zero whenever the group is well chosen. Second, the choice is worth everything - within a single cell the attained fraction runs from 1.00 down to 0.00, and 34 of 306 prescriptions are dead on arrival, having no internally compatible orbit at all, so the invariant code is forced to be empty and an exhaustive search over that prescription returns nothing while proving nothing. Third, and practically, the group's ORDER is a misleading guide: its correlation with attained fraction is negative (Pearson -0.527, Spearman -0.533), and attainment is not monotone in order. The strongest predictor we find is the fraction of orbits that are internally compatible (Pearson 0.559, Spearman 0.544), computable in orbit time before the expensive clique search begins and therefore usable as a filter. Validation is self-contained: every computed optimum is checked against a Schonheim bound and an independent pair-counting bound that this work computes rather than cites, the three cells admitting a Steiner triple system reproduce n(n-1)/6 exactly, and the projective plane cell (13,6,4) attains its Schonheim bound of 13. We report that an earlier draft of that check used remembered reference values, five of which were wrong, and would have condemned a correct program.