Extremal 72

Extremal 72 is a community effort to decide whether a binary Type II [72,36,16] code exists.

The project has two possible endings:

• Find the code, producing a remarkable extremal self-dual object connected to designs, lattices, quantum CSS codes, and code/CFT constructions.
• Rule it out, settling a problem posed by Sloane in 1973.

Start Here

• Current Map: the short status view.
• The Hierarchy: the 72 -> 56 -> 40 -> 24 strategy.
• Length-40 Menu: the current finite menu and MENU stats.
• Tests: one page for each test T1 through T32.
• Weight Enumerators: downloadable enumerator data and the triweight routine bundle.
• Validation Standards: exactness requirements for public eliminations and downloadable computations.
• Ideas Not Pursued: useful negative results and parked directions.

Public Status

The project is open. The current public headline is:

132 possible length-40 shadows
60 eliminated by exact obstructions
72 still viable at the menu level
51 witnessed nonempty
21 not yet exhausted

A surviving menu row is not a code. It is a still-compatible shadow that a hypothetical [72,36,16] code could cast.