Extremal72

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 [72,36,16] code. A construction would feed directly into related objects such as a self-dual quantum CSS code and conformal-field-theory data; a nonexistence proof would settle a long-standing coding-theory problem.

How the search works: rather than sift through the astronomically many length-72 codes directly, we reason about their weight enumerators — the counts of codewords at each weight — and the finite list of arithmetic shadows those counts can take. Anchoring at a minimum-weight codeword and projecting down a residual tower, [72] → [56] → [40] → [24], forces each shorter descendant to carry a specific enumerator; computing these anchored projections, including their higher-genus (bi- and tri-weight) forms, squeezes out the constraints that decide a branch. When no code can meet the forced arithmetic the branch is ruled out, while explicitly building one up the tower would settle existence — so every result here is either an exact obstruction or a witnessed descendant.

Current public posture: 72 compatible shadows remain. 51 have witnessed nonempty descendants; 21 are still unresolved as existence questions.

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 hypothetical [72,36,16] code to one of just five: C1 (trivial), C2, C3, C2×C2, or C5. Excluded along the way: order 2 with fixed points (Bouyuklieva 2002); Z7, Z32, D10 (Feulner–Nebe 2011); C8, Q8, Z4×Z2, Z10 (Nebe 2012); order 6 (Borello 2012); C4 (Yorgov–Yorgov 2013); S3, A4, D8 (Borello–Dalla Volta–Nebe 2013); C23 (Borello 2014); with the short list and solvability consolidated by O'Brien–Willems (2011) and Bouyuklieva–O'Brien–Willems (2024). Full citations are in the References catalogue.

We make no automorphism assumption. The trivial group C1 imposes no structure — every codeword orbit has size one — so it is the hardest case to rule out, and every test and enumerator on this site is automorphism-agnostic: it must hold irrespective of any symmetry. The narrowed list drives a parallel prescribed-automorphism search; 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 by a proven map:

A 5-(72,16,78) design. The 249849 weight-16 codewords form a 5-design: every 5 coordinates sit together in exactly 78 of them. Why: the Assmus–Mattson theorem applied to the extremal Type II code (minimum weight 16, dual distance 16) makes each weight class a 5-design, and counting fixes λ = 249849·C(16,5)/C(72,5) = 78.

A code CFT at central charge c = 36. The code maps to a chiral conformal field theory of central charge c = n/2 = 36. Why: in the code–CFT dictionary (Henriksson–Kakkar–McPeak, arXiv:2112.05168) a length-n doubly-even self-dual code yields a chiral CFT of central charge n/2, whose genus-g partition function is the theta lift Θ of the genus-g weight enumerator — a degree-g, weight-18 Siegel modular form. This site computes that genus-3 Θ-projection (see the Enumerators tab).

A [[71,1,≥15]] self-dual CSS code. Puncturing the self-dual [72,36,16] code in one coordinate and using the punctured dual C⊥ as both the X- and Z-stabilizer gives a self-dual CSS code (CX = CZ) with parameters [[71,1,≥15]]. Why: puncturing gives C = [71,36,≥15] with C⊥ ⊆ C, so CSS(C⊥,C⊥) is valid with k = 71 − 2·35 = 1 and distance ≥ d−1 = 15 (Jain–Albert, arXiv:2408.12752).