The Length-40 Menu

The length-40 menu is the public ledger of the residual search.

Every hypothetical [72,36,16] Type II code casts one of these length-40 shadows:

E = [40,k,>=16], doubly-even, self-orthogonal, 1_40 in E.

The weight enumerator is:

W_E(y; a,b) = 1 + a(y^16 + y^24) + b y^20 + y^40.

Current Counts

132 raw candidates
60 eliminated by exact obstructions
72 surviving at the menu level
51 witnessed nonempty
21 not yet exhausted

How To Read These Numbers

• 132 is the complete arithmetic menu from the current exact validity filter.
• 60 rows have proof-grade eliminations.
• 72 rows remain viable as shadows.
• 51 surviving rows have explicit length-40 witnesses.
• 21 rows are unresolved: not witnessed and not fully exhausted.

A surviving row is not a [72,36,16] code. It is a compatible shadow.

C_5 Sub-Menu (Automorphism-Conditional Tag)

The menu makes no assumption about the code's automorphism group. A series of papers narrows that group to one of five — C_1 (trivial), C_2, C_3, C_2×C_2, or C_5 — but the trivial group C_1 imposes no structure and is the hardest case, so every test here must hold for it. (See the Overview for the exclusion citations.)

One branch still leaves a clean fingerprint on the menu. For C_5, orbit counting (A_16 = 249849 ≡ 4 and the 5082 ≡ 2 (mod 5) residual partners) forces a σ-fixed disjoint anchor pair whose length-40 residual carries a fixed-point-free C_5 action, so its menu row satisfies a ≡ 0 (mod 5).

So a C_5-symmetric code must realize one of these 16 surviving rows (a ≡ 0 mod 5; 14 core rows plus the two reinstated (7,15,96), (8,55,144)):

k=6:  (5,52)   (15,32)   (25,12)
k=7:  (15,96)  (25,76)   (35,56)  (45,36)  (55,16)
k=8:  (55,144) (75,104)  (95,64)  (115,24)
k=9:  (135,240)(175,160) (215,80)
k=10: (295,432)

What the tag means. If automorphism-agnostic tests eliminate all of these, the C_5 branch closes with no Hermitian F_16 search — a far cheaper target than killing the whole menu. The converse does not hold: ruling out C_5 does not remove these rows, since a trivial-automorphism code could still cast any of them. The tag marks where the C_5 branch overlaps the menu; it is not a kill.