Searching for a [72,36,16] extremal code
Public mission
A crowd search for the missing extremal Type II code
The project tries to find or rule out a binary doubly-even self-dual<br>[72,36,16] code. A construction would feed directly into related<br>objects such as a self-dual quantum CSS code and conformal-field-theory<br>data; a nonexistence proof would settle a long-standing coding-theory<br>problem.
How the search works: rather than sift through the<br>astronomically many length-72 codes directly, we reason about their<br>weight enumerators — the counts of codewords at each weight<br>— and the finite list of arithmetic shadows those counts<br>can take. Anchoring at a minimum-weight codeword and projecting down a<br>residual tower, [72] → [56] → [40] → [24],<br>forces each shorter descendant to carry a specific enumerator;<br>computing these anchored projections, including their higher-genus<br>(bi- and tri-weight) forms, squeezes out the constraints that decide a<br>branch. When no code can meet the forced arithmetic the branch is ruled<br>out, while explicitly building one up the tower would settle existence<br>— so every result here is either an exact obstruction or a witnessed<br>descendant.
Current public posture:<br>72 compatible shadows remain. 51 have witnessed nonempty descendants;<br>21 are still unresolved as existence questions.
132<br>raw MENU candidates
60<br>proof-grade eliminations
72<br>surviving shadows
21<br>unresolved rows
Why this is a good crowd problem
Many doors: algebra, coding theory, design theory, SDP, exact enumeration, and computational proof can all contribute.
Finite checkpoints: every test has a page, an input menu row, a status, and a way to reproduce or improve the obstruction.
Useful outcomes on both sides: construction gives new highly structured objects; impossibility resolves the length-72 extremal question.
Automorphism group: narrowed, but not assumed
A long series of papers has reduced the possible automorphism group of a<br>hypothetical [72,36,16] code to one of just five:<br>C1 (trivial), C2 ,<br>C3 , C2×C2 ,<br>or C5 . Excluded along the way:<br>order 2 with fixed points (Bouyuklieva 2002);<br>Z7, Z32, D10 (Feulner–Nebe 2011);<br>C8, Q8, Z4×Z2, Z10 (Nebe 2012);<br>order 6 (Borello 2012); C4 (Yorgov–Yorgov 2013);<br>S3, A4, D8 (Borello–Dalla Volta–Nebe 2013);<br>C23 (Borello 2014); with the short list and solvability<br>consolidated by O'Brien–Willems (2011) and<br>Bouyuklieva–O'Brien–Willems (2024). Full citations are in the<br>References catalogue.
We make no automorphism assumption. The trivial group<br>C1 imposes no structure — every codeword orbit has size one<br>— so it is the hardest case to rule out, and every test and enumerator<br>on this site is automorphism-agnostic: it must hold irrespective of any<br>symmetry. The narrowed list drives a parallel prescribed-automorphism search;<br>it is not an assumption the menu relies on.
If the code is found, these structures come with it
A construction is not an isolated object — three structures follow, each<br>by a proven map:
A 5-(72,16,78) design. The 249849 weight-16 codewords form<br>a 5-design: every 5 coordinates sit together in exactly 78 of them.<br>Why: the Assmus–Mattson theorem applied to the extremal Type II<br>code (minimum weight 16, dual distance 16) makes each weight class a<br>5-design, and counting fixes<br>λ = 249849·C(16,5)/C(72,5) = 78.
A code CFT at central charge c = 36. The code maps to a<br>chiral conformal field theory of central charge c = n/2 = 36.<br>Why: in the code–CFT dictionary (Henriksson–Kakkar–McPeak,<br>arXiv:2112.05168) a length-n doubly-even self-dual code yields a chiral CFT<br>of central charge n/2, whose genus-g partition function is the theta lift<br>Θ of the genus-g weight enumerator — a degree-g, weight-18 Siegel<br>modular form. This site computes that genus-3 Θ-projection (see the<br>Enumerators tab).
A [[71,1,≥15]] self-dual CSS code. Puncturing the<br>self-dual [72,36,16] code in one coordinate and using the punctured dual<br>C⊥ as both the X- and Z-stabilizer gives a self-dual CSS<br>code (CX = CZ) with parameters [[71,1,≥15]].<br>Why: puncturing gives C = [71,36,≥15] with<br>C⊥ ⊆ C, so CSS(C⊥,C⊥) is<br>valid with k = 71 − 2·35 = 1 and distance ≥ d−1 = 15<br>(Jain–Albert, arXiv:2408.12752).
Main route
The 72 -> 56 -> 40 -> 24 hierarchy
The public story should center this descent: start from a hypothetical<br>[72,36,16] Type II code, take a weight-16 word, study the length-56 residual,<br>descend through length-40 menus, and finally reach length-24 endpoint tests.
Start<br>[72,36,16]<br>Type II code, A_16 = 249849
Residual<br>[56,21,16]<br>5082 minimum words forced
Menu<br>[40,k,16]<br>132 shadows, 72 surviving
Endpoint<br>[24,1,24]<br>local tests and exhaustions
What the hierarchy buys
The descent turns a global code-existence question into a finite set of<br>compatible length-40 shadows. Each row can then be attacked by exact<br>divisibility, local gluing, support-weight constraints, SDP layers, or<br>direct existence searches.
Forced...