Signed difference sets from perfect nonlinear functions
Abstract
A signed difference set in a finite abelian group G of order v is an element D=Σ_(g)a_g g of ℤ[G] with a_g ∈ {0, ± 1}, support size k, satisfying DD^((-1))=(k-λ) 0_G+λ G. He and Wu recently produced a (125,28,3) signed difference set in C₅³ out of quartic multiplicative characters of 𝔽₂₅ and 𝔽₅, together with quartic Jacobi sums. We show that their signed set is, verbatim, the graph of the norm map Nm:𝔽₂₅ → 𝔽₅ with the subgroup {0} × 𝔽₅ subtracted, so that no character theory is needed to see it. What that identity uses is only perfect nonlinearity: if f: A → B is a map of finite abelian groups with |A|=n², |B|=n and every nonzero derivative of f balanced, then Γ_f-({0} × B) is an (n³, n²+n-2, n-2) signed difference set. Binary quadratic forms of nonzero discriminant supply such an f over every finite field and in every characteristic, so there is a signed difference set with parameters (q³, q²+q-2, q-2) in the elementary abelian group of order q³, for every prime power q; the He–Wu example is the case q=5. Three members of the family — at (27,10,1) in C₃³, (64,18,2) in C₂⁶ and (343,54,5) in C₇³ — occupy cells recorded as open in Gordon's signed difference set database, and two further members are beyond its parameter range. We also exhibit two inequivalent (125,28,3) signed difference sets in C₅³, separated by an affine-covariant invariant, so the He–Wu set is one of at least two classes there. All statements are formally verified in Lean 4.
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Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
File fingerprint
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Claim ledger
Stated results
SD1known2026-08-29
The principal-character equation (|P|-|N|)² = k + lambda(v-1) for signed difference sets, proved coefficientwise over any finite abelian group
SD2routine2026-08-29
Every (125,28,3)-SDS in an abelian group has |P|,|N| = 24,4
SD3routine2026-08-29
No (125,28,3)-SDS has all coefficients >= 0: the cell is unreachable by an ordinary difference set
SD4candidate2026-08-29
The graph of a perfect nonlinear map minus its forbidden subgroup is a signed difference set: |A| = n², |B| = n, every nonzero derivative of f: A -> B balanced implies Gamma_f - (0 x B) is an (n³, n²+n-2, n-2)-SDS
SD5candidate2026-08-29
The infinite family: over any finite field F with |F| = q, any binary quadratic form of nonzero discriminant gives a (q³, q²+q-2, q-2)-SDS in Cₚ³ᵐ
SD6known data2026-08-29
The (125,28,3)-SDS in C₅³ as the q = 5 member of the family, from the norm form X² - 2Y² of F₂5/F₅
SD7candidate2026-08-29
Five further members of the family certified: (27,10,1) in C₃³, (64,18,2) in C₂⁶, (343,54,5) in C₇³, (1331,130,9) in C₁1³, (2197,180,11) in C₁3³
SD8routine2026-08-29
Controls for the family: the degenerate discriminant fails at the same k = 28; adding or deleting one support point is refuted for every parameter triple; v, k, lambda are pinned
SD9routine2026-08-29
Arithmetic rigidity of the He-Wu ansatz: d(q+1) = d(e+1)+1 with q = ed+1 forces d = 1, i.e. the character order must be the full q-1
SD10known data2026-08-29
The quartic Jacobi-sum data of He-Wu's Lemma: multiplicity tables (2,7,4,8,4) and (2,0,1,2,0), J(rho,rho) = 3+4i, J(sigma,sigma) = -1+2i, J(rho,rho) conj J(sigma,sigma) = 5-10i
SD11routine2026-08-29
He-Wu's character-defined signed set is exactly the graph of the norm map N: F₂5 -> F₅ minus the subgroup 0 x F₅
SD12known data2026-08-29
He-Wu Theorem 5.6 certified: their D = P - N from quartic characters is a (125,28,3)-SDS in C₅³, with |P| = 24 and |N| = 4
SD13known data2026-08-29
He-Wu Example 5.4 certified: the Desarguesian-spread signed sets SDS(64,22,6) in C₂⁶ and SDS(256,106,42) in C₂⁸, with (|P|,|N|) = (21,1) and (105,1)
SD14routine2026-08-29
An affine map matching two signed difference sets has injective linear part, whenever k!= lambda
SD15candidate2026-08-29
There are at least two inequivalent (125,28,3)-SDSs in C₅³: the He-Wu (elliptic, norm-form) one and a hyperbolic one, separated by an affine-covariant invariant
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Let G be a finite abelian group of order v, written additively, and identify a subset of G with the corresponding element of the integral group ring Z[G]. A signed subset is an element
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7