A new public research project is inviting mathematicians, programmers, and curious hackers to join the search for something that's eluded experts for decades: a binary doubly-even self-dual [72,36,16] code. The mission statement is refreshingly direct—either construct this extremal Type II code or prove it doesn't exist. Either outcome advances the field. The project website at valbert4.github.io/selfdual_site/ lays out the computational strategy and current status for anyone ready to contribute.

How the Search Works

Brute-forcing all length-72 codes is astronomically impossible, so the team takes a clever structural approach using weight enumerators—counts of codewords at each weight—and their arithmetic shadows. The method anchors at a minimum-weight codeword and projects down through a tower: [72] → [56] → [40] → [24]. Each shorter descendant must carry a specific enumerator, which squeezes out constraints that decide whether a branch can produce a valid code. When no code satisfies the forced arithmetic, that branch is ruled out. Explicitly building one up would settle existence outright.

Current Progress: 21 Shadows Left to Rule Out

As of late June 2026, 72 compatible shadows remain in play. Of those, 51 have witnessed nonempty descendants—meaning viable candidate structures are still alive in those branches. But 21 shadows remain unresolved as genuine existence questions. Every result from this process is either an exact obstruction (ruling out that branch) or a witnessed descendant (a code-like structure that might lead somewhere). The team isn't just hoping for construction; they're methodically eliminating the search space one constraint at a time.

Why This Is a Crowd Problem Worth Solving

The project page makes the case directly: multiple doors into this problem exist. Algebra, coding theory, design theory, semidefinite programming (SDP), exact enumeration, and computational proof can all contribute. Every test has a dedicated page, an input menu row, status tracking, and reproducibility built in—you can reproduce or improve any obstruction. And the outcomes cut both ways: construction yields new highly structured objects; impossibility resolves the length-72 extremal question for good.

The Automorphism Angle

A long series of papers dating back to Bouyuklieva (2002) has progressively narrowed the possible automorphism group of a hypothetical [72,36,16] code. The current candidate list is down to just five: C1 (trivial), C2, C3, C2×C2, or C5. Excluded groups include order-2 with fixed points, Z7, Z32, D10, C8, Q8, Z4×Z2, S3, A4, and many others via Feulner–Nebe (2011), Nebe (2012), Borello (2012), Yorgov–Yorgov (2013), Borello–Dalla Volta–Nebe (2013), and Borello (2014). Critically, the project makes no automorphism assumption. The trivial group C1 imposes zero structure—every codeword orbit has size one—so it represents the hardest case to eliminate. All tests on this site are automorphism-agnostic by design.

What Comes With a Solution

If someone finds the code, three major structures follow automatically via proven mathematical maps: First, a 5-(72,16,78) design where the 249,849 weight-16 codewords form a combinatorial structure—every 5 coordinates sit together in exactly 78 of them. Second, a chiral conformal field theory at central charge c = 36, mapping to string theory via the theta lift of genus-g weight enumerators. Third, a [[71,1,≥15]] self-dual CSS code for quantum error correction, derived by puncturing one coordinate and using the dual as both X- and Z-stabilizers.

Key Takeaways

  • The search uses weight enumerator constraints rather than brute force, projecting through towers: [72] → [56] → [40] → [24]
  • 21 of 72 compatible shadows remain unresolved; 51 have viable descendant structures that need further analysis
  • No automorphism assumptions are made—the trivial group case is the hardest to rule out and must be handled directly
  • A construction unlocks a 5-design, CFT at c=36, and quantum error-correcting code simultaneously

The Bottom Line

This is the kind of problem that makes you appreciate how much heavy lifting happens in pure mathematics before applications emerge—finding or ruling out this one code would cascade into quantum computing and theoretical physics. If you've got the algebra chops or just want to contribute compute cycles, the checkpoint system on the site lets you reproduce existing results while potentially carving out your own corner of a decades-old open problem.